\documentstyle[12pt]{article}
\begin{document}
%DEFINITION OF THE MACROS USED
\newcommand{\C}{C\!\!\!\rule[.5pt]{.7pt}{6.5pt}\:\:}
\newcommand{\R}{I\!\!R}
\newcommand{\N}{I\!\!N}
\newcommand{\Z}{Z\!\!\!Z}
\newcommand{\BB}{{\cal B}}
\newcommand{\DD}{{\cal D}}
\newcommand{\HH}{{\cal H}}
\newcommand{\NN}{{\cal N}}
\newcommand{\OO}{{\cal O}}
%%END OF THE DEFINITION%%
\begin{center}
{\Huge\bf A list of abstracts}
\end{center}
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Invited talks:}
\vspace{10mm}
\begin{center}
{\Large\bf Quantum adiabatic theorems} \\ [5mm]
{\large J.E.~Avron} \\ [2mm]
Department of Physics, Technion, 32000 Haifa, Israel
\end{center}
\noindent
The usual adiabatic theorems in Quantum Mechanics posits the
existence of a gap in the spectrum. We developed a strategy for
studying adiabatic theorems in quantum mechanics which avoids an
explicit gap condition and has the flavor of the Riemann-Lebesgue
lemma. I will illustrate this method for the study of the following
problem: In QED, photons tend to lead to gapless spectrum. This
raises the question why, then, is the adiabatic theorem of quantum
mechanics which disregards photon so successful? To understand the
role of soft photons we prove an adiabatic theorem for the ground
state of a simple model Hamiltonians that allows soft photons. We
show that for weak electron-photon coupling, the adiabatic time scale
is close to the time scale in a theory without the quantized
radiation field up to a Lamb shift. The inclusion of photons leads to
a logarithmic correction in three dimensions coming from an infrared
singularity characteristic of QED. The results have been obtained
in collaboration with A.~Elgart.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Electronic transport revisited:\\ the variable range
hopping theory} \\ [5mm]
{\large J.~Bellissard} \\ [2mm]
UMR 5626, CNRS and Laboratoire de Physique Quantique, \\
Universit\'e Paul-Sabatier, 118, Route de Narbonne, F--31062 Toulouse
\end{center}
\noindent
The purpose of this talk is to give a mathematical model, based upon
a quantum version of the kinetic theory, liable to describe the Mott
hopping transport in lightly doped compensated semiconductors at very
low temperature. The physical framework will be given first. The
Mott argument leading to the conductivity will be explained.
Then a simple Drude-like kinetic quantum model will be given within
the Relaxation Time Approximation (RTA). Several results like the
proof of the validity of Kubo's formula, the anomalous Drude formula
will be given. Then the model will be modified to take into account
the low temperature physics. The latest rigorous results about it
will be given together with comments about consequences on the Kubo
formula and the Mott estimate.
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Two--dimensional periodic magnetic Hamiltonian with
discontinuous vector potential is absolutely continuous} \\ [5mm]
{\large M.S.~Birman} \\ [2mm]
Institute of Physics, Sankt Peterburg State University, \\
Ulyanovskaya 1--1, 198804 Sankt Peterburg, Russia
\end{center}
\noindent
The periodic operators of mathematical physics are discussed. The
main question is the absolute continuity of the spectrum. We propose
the survey of the obtained results and the unsolved problems in this
sphere. Recently the author (jointly with T.Suslina) proved the
absolute continuity of the spectrum of twodimensional periodic
magnetic Hamiltonian with discontinuous potentials. Precisely, it is
assumed that a magnetic potential $A\in L_r (\Omega),\; r>2$, and an
electric potential $V\in L_\rho (\Omega),\; \rho >1\;$ (here $\Omega$
is the elementary cell of the lattice of periods).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Quantum nonergodic behaviour} \\ [5mm]
{\large G.~Casati} \\ [2mm]
International Center for the study of dynamical systems,
University of Milano at Como, Via Lucini, 3 - 22100 Como, Italy;
{\em Casati@fis.unico.it}
\end{center}
\noindent
We discuss the phenomenon of quantum localization in conservative,
classically chaotic, dynamical systems. This phenomenon implies
statistical relaxation to a non ergodic quantum steady state and
deviations from the predictions of random matrix theory.
As concrete examples we discuss 2D quantum billiards.
For classically chaotic open systems, we show the appearance
of quantum fractal sets and that the quantum
relaxation process is characterized by a new quantum relaxation time scale
which is much shorter than the Heisenberg time and much larger than the
Ehrenfest time.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Some spectral problems related \\ to photonic crystals} \\
[5mm]
{\large J.--M.~Combes} \\ [2mm]
Departement de math\'{e}matiques, Universit\'{e} de Toulon et du Var,
\\ F--83957 La Garde Cedex, France
\end{center}
\noindent
Photonic crystals are a source of new and important challenges for
Mathematical Physics. Among them is the existence of gaps in the
energy spectrum of electromagnetic waves propagating in a periodic
dielectric structure; we present some of the recent analytical and
numerical existence results obtained by different authors in high
contrast media. Randomization of the periodic structure is then shown
to lead in some regimes to Anderson localization of light.
Radiative decay of excited atoms in such systems is strongly
inhibited if the transition frequency falls into a gap or at band
edges where localisation holds. Discussion of this effect requires
in particular quantization of electromagnetic field in non
homogeneous media as developed recently by A.~Tip .
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Uniqueness theorems for Schr\"odinger, Jacobi and
Dirac--type operators and recovery of potentials for various spectral
data} \\ [5mm]
{\large F.~Gesztesy} \\ [2mm]
Department of Mathematics, University of Missouri \\
Columbia, MO 65211, USA
\end{center}
\noindent
We plan to review a number of recent uniqueness results in the
inverse spectral theory for one-dimensional Schr\"odinger, Jacobi,
and Dirac-type operators, and possibly describe some extensions to
the case of matrix-valued coefficients. In particular, we intend
to describe a new circle of inverse spectral problems related to
recovering the potential from partial knowledge of the potential
(on an appropriate subinterval) and partial spectral information.
The principal tools involved are Weyl-Titchmarsh functions,
densities of zeros of certain classes of entire functions, and
trace formulas.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf The ground state of low density Bose gas} \\ [5mm]
{\large E.H.~Lieb} \\ [2mm]
Princeton University, Jadwin Hall, Princeton, NJ 08544--0708, U.S.A.
\end{center}
\noindent
Now that the properties of low temperature Bose gases at low density,
$\rho$, can be examined experimentally it is appropriate to revisit some
of the formulas deduced by many authors 4-5 decades ago. One of these is
that the leading term in the energy/particle is $2\pi \hbar^2 \rho a/m$,
where $a$ is the scattering length. Owing to the delicate and peculiar
nature of bosonic correlations, four decades of research had failed to
establish this plausible formula rigorously. The only previous lower
bound for the energy was found by Dyson in 1957, but it was 14 times too
small. The correct asymptotic formula has very recently been obtained
jointly with J. Yngvason and this work will be presented here. Another
problem of great interest is the existence of Bose-Einstein condensation,
and what little is known about this rigorously will also be discussed.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf On a heuristic derivation of the Mott formula for low
frequency conductivity of disordered systems} \\ [5mm]
{\large L.A.~Pastur} \\ [2mm]
Mathematique, Universit\'{e} Paris VII, 2, place Jussieu, F--75252
Paris
\end{center}
\noindent
We present a version of the derivation of the low frequency
asymptotics of the ac conductivity of a disordered system in the
strong localization regime of the one-body approximation. The form of
this asymptotics was proposed by N.~Mott in the late 60's. Our
analysis is based on the hypothesis (the ansatz) that in the strong
localization regime relevant realizations of the random potential
have the form of of broad and/or deep potential wells that are
uniformly and chaotically distributed in the space, can be
parameterized by at least one continuously distributed parameter and
have small density. Since this ansatz is accepted, we can the use
rather general, simple and unambiguous (at least on the formal level)
means: the concentration expansion, the analysis of the tunnelling in
the system of several wells, etc. We show first that the density of
wells coincides in the leading order with the density of states in a
given small interval of energies and that because of this the the
density of states is in fact the small parameter of the theory in the
strong localization regime. This picture of the strong localization
regime was proposed by I.~Lifshitz in the early sixties. Then we
derive the Mott formula and a number of other similar formulas for
various physical and spectral characteristics of disordered systems.
\vspace{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Integrals of the discrete quantum pendulum} \\ [5mm]
{\large R.~Seiler} \\ [2mm]
FB Mathematik der TU Berlin, Stra{\ss}e des 17 Juni 136, D-1000 Berlin
12
\end{center}
\noindent
Discrete surfaces of constant negative curvature are parametrized by
solutions of the discrete sine-Gordon equation (Bobenko and Pinkall).
The so defined dynamical systems is completely integrable. Quantization
leads to operators of the Hofstadter type. Their spectrum can be
investigated by the Bethe Ansatz.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf A new approach to inverse spectral theory} \\ [5mm]
{\large B.~Simon} \\ [2mm]
Mathematics 253--37, California Institute of Technology, \\ Pasadena,
CA 91125, U.S.A.
\end{center}
\noindent
A new approach to inverse spectral theory for 1D Schrodinger
operators will be presented. Just as the Gel'fand Levitan theory is
the analog of the orthogonal polynomial approach to inverse spectral
theory for Jacobi matrices, this new approach is the analog of the
continued fraction approach to the Jacobi problem. The approach
involves and implies a new understanding of the asymptotics of the
Weyl m function as the energy goes to minus infinity. An interesting
new non-linear equation is part of the picture.
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Quantum dots} \\ [5mm]
{\large J.~Yngvason} \\ [2mm]
Institut f\"ur Theoretische Physik der Universit\"at Wien, \\
Boltzmanngasse 5, A--1080 Wien
\end{center}
\noindent
Quantum dots are mesoscopic systems consisting of electrons confined
to a small region within a semiconductor heterostructure. The motion
of the electrons is essentially restricted to two dimensions and
their number may vary from a few up to several thousand or even
millions. Quantum dots can in many respect be regarded as artificial,
two dimensional atoms, but the effective parameters are tunable to a
certain extent and may differ appreciately from their natural
counterparts.Important differences arise also because the Coulomb
interaction between the electrons is not harmonic in two dimensions
and because the confining potential behaves differently from the
potential of an atomic nucleus at small and large distances from the
center. Quantum dots can be studied by various spectroscopical means
(optical, charge tunneling...), and they provide an experimental and
theoretical testing ground for quantum mechanics. Besides, they have
potential technical applications as nanoscale electronic devices. The
lecture will bring a review of rigorous results in the theory of
quantum dots.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section*{Plenary talks:}
\vspace{8mm}
\begin{center}
{\Large\bf Singular Bohr--Sommerfeld \\ quantization condition} \\ [5mm]
{\large Y.~Colin de Verdiere} \\ [2mm]
Institut Fourier, F-38402 Saint Martin d'Heres Cedex
\end{center}
\noindent
We will describe the extension of Bohr-Sommerfeld quantization
conditions to the case of Morse singularities for the classical
limit of a completely integrable semi-classical Hamiltonian system.
Our first example (in collaboration with Bernard PARISSE) is a 1d
Schr{\"o}dinger operator with a symmetric double well were we are
able to describe the disappearance of the parity pair when crossing
the critical energy. Our second example (worked out by San NGOC VU)
is the focus-focus singularity which is obtained for a pendulus
inside a linear potential in the 2-sphere.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Optimizing eigenvalues of Schr\"odinger operators with
potentials depending on curvature} \\ [5mm]
{\large \underline{E.M.~Harrell II}, M.~Loss} \\ [2mm]
School of Mathematics, Georgia Institute of Technology, Atlanta
\end{center}
\noindent
We estimate and optimize eigenvalues of Schr\"odinger operators
of the form $T+Q(k)$ defined on curves or manifolds. Here the
kinetic energy T is the negative of the Laplace-Beltrami operator
and $Q$ is a quadratic expression in the curvatures. Such operators
arise in studies of quantum waveguides and of metal alloys. For
various $Q$'s we determine the manifolds which optimize the first and
second eigenvalues.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Spectral properties of strongly inhomogeneous periodic
media} \\ [5mm]
{\large R.~Hempel} \\ [2mm]
Institut f\"ur Analysis, TU Brauschweig
\end{center}
\noindent
We investigate the band-gap structure and the
integrated density of states for periodic divergence type operators
$$
T_\lambda = - \nabla^* ( 1 + \lambda \chi_\Omega) \nabla,
$$
in the limit $\lambda \to \infty$,
where $\Omega$ is an open, periodic subset of ${\bf R}^m$;
we also assume
that the complement of $\Omega$ does not intersect the boundary of
the fundamental period cell. Operators similar to $T_\lambda$
might occur in a
simple model for heat conduction in a metal with air bubbles.
Among other results, we find that the $k$-th spectral gap for
$T_\lambda$ will be open for $\lambda$ large
if and only if the $k$-th Dirichlet
eigenfunction on $M_0$ has a non-zero integral,
where $M_0$ is the part of $\Omega^C$ contained inside the fundamental
period cell. While it is well-known that Dirichlet eigenvalues
are generically simple (considered as functions of the underlying
domain), the property required here is more subtle and we have only
partial results on genericity. (Joint work with K.~Lienau, Braunschweig.)
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Some recent results in time--dependent Schr\"odinger
operators} \\ [5mm]
{\large A.~Jensen} \\ [2mm]
Department of Mathematics, Institute for Electronic Systems \\
Aalborg University, Fr. Bajers Vej 7E, DK-9220 Aalborg Oest, Denmark
\end{center}
\noindent
We consider a Schr\"{o}dinger operator $H(t)=-\Delta+V(t)$ with a real
time-dependent potential. The space-time scattering theory is studied
in spaces $L^r({\R};L^q({\R}^d))$ for a certain range of $r,q$. A new
representation formula for the scattering operator is found.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf ${\cal H}_{-2}$--construction of general type and some
applications to the spectral theory} \\ [5mm]
{\large \underline{S.T.~Kuroda}, H.~Nagatani} \\ [2mm]
Department of Mathematics, Gakushuin University, Tokyo
\end{center}
\noindent
The ${\cal H}_{-2}$--construction is a way of defining the perturbed
operator formally given as $\,H=H_0\!+V\,$ when $\,V\,$ is highly
singular with respect to $\,H_0\,$. (An example is $\,H= -\,{d^2\over
dx^2}\, +\,\alpha (\cdot,\delta')\delta'(x)\,$.) A.~Kiselev and
B.~Simon (J. Funct. Anal. {\bf 130} (1995), 345-356) introduced this
construction for rank one perturbations related to $\,V=c\langle
\cdot,\varphi \rangle\varphi\,$. In ${\cal H}_{-2}$--construction one
supposes the situation that $\,\varphi\,$ is not possibly in the
basic Hilbert space $\,\HH_0$ but only in ${\cal H}_{-2}=\DD(H_0)^*$,
the dual of the domain ${\cal H}_{2}=\DD(H_0)$ of with the graph norm.
Our ${\cal H}_{-2}$--construction is an extreme generalization of
this rank one construction. Recall that given two bounded selfadjoint
operators $\,H_0$ and $\,H\,$, they are related as $\,H=H_0\!+V\,$,
where $\,V\,$ is a bounded selfadjoint operator. Likewise, we shall
see that any two selfadjoint operators (not necessarily bounded) can
be related through our ${\cal H}_{-2}$--construction via resolvents.
For a fixed $\,H_0$ this construction will serve to classify all
selfadjoint operators in terms of the regularity (or singularity)
with respect to $\,H_0$. In particular, we can characterize when
$\,H\,$ is a selfadjoint extension of $\,H_0|_\NN$, the restriction
of $\,H_0$ to$\,\NN\subset \DD(H_0)\,$. We shall first present a
general theory and then discuss two examples: (i) a complete
description of one-dimensional point interactions; (ii) one
particular example of surface interaction in three dimension. Our
theory also includes the case when $\,H\,$ is not selfadjoint.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Scattering with time periodic potentials \\ and cyclic
states} \\ [5mm]
{\large Ph.-A.~Martin} \\ [2mm]
Institut de physique th\'eorique, Ecole Polytechnique F\'ed\'erale \\
CH--1015 Lausanne, Switzerland
\end{center}
\noindent
We consider a scattering system $H=H_0+V(t)$ with free Hamiltonian
$H_0$ having an absolutely continuous spectrum in the band $[0,E_0]$
and $V(t)$ is a time-periodic potential with period $T$. We show that
the corresponding monodromy operator has in general both absolutely
continuous spectrum and point spectrum (cyclic states) when the
period is small enough. In fact, these cyclic states originate from
the eigenstates of the static time-averaged Hamiltonian
$\overline{H}= {1\over T}\, \int_0^T dt\,H(t)\:$: they lose their
stability as $T$ increases and may be transformed into resonances.
These mechanisms are illustrated on the explicitly solvable model of
a periodically kicked rank-one perturbation.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Geometry of decay properties \\ for resolvent of a
self--adjoint operator} \\ [5mm]
{\large E.~Mourre} \\ [2mm]
Centre de Physique Th\'eorique, CNRS, Marseille Luminy
\end{center}
\noindent
For a class of self-adjoint operators we show how to estimate the
decay of their resolvent at a fixed energy by using the spectral or
stability distance; for one of them we analyze the behaviour of decay
properties under scaling.
In the case of an ergodic family of one-dimensional operators, we
show the equivalence between the positivity of the Lyapounov exponent
and the asymptotic proportionality of stability distances to the
Euclidean ones.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Boundary condition dependence for spectral measures of
Sturm--Liouville operator} \\ [5mm]
{\large R.~del Rio Castillo} \\ [2mm]
Instituto de Investigaciones en Mathematicas Aplicadas, \\
Universidad Nacional Autonoma de Mexico, \\ AP 20--726 Deleg.
A.~Obregon, 01000 Mexico, D.F.
\end{center}
\noindent
We study the behavior of spectra of ordinary differential
operators when the boundary condition changes. It is shown that
there exists a small set (first category) which is a common support
for all the spectral measures corresponding to Sturm-Liouville
Operators on the half axis. It is also proven that coexistence of
absolutely continuous and singular continuous spectrum is possible.
This is joint work with Alexei Poltoratski.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section*{Session talks:}
\vspace{8mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf The anisotropic Aharonov-Bohm effect}
\\ [5mm]
{\large R.~Adami} \\ [2mm]
Dipartimento di Matematica``Castelnuovo" , Universita' di Roma \\
``La Sapienza", Piazzale Aldo Moro 5, 00185 Roma, Italy
\end{center}
\noindent
We study the interaction of a charged spinless non relativistic
particle with an ideal solenoid as the superposition of a purely
magnetic interaction (namely, the Aharonov-Bohm effect) with a
pointwise one, describing the contact interaction with the actual
support of the magnetic field. We use the formalism of first
quantization and the von Neumann-Krein theory for the construction of
self-adjoint extensions for symmetric operators. Finally we compute
the scattering amplitude for all the obtained hamiltonians. The main
physical result we achieve is that in spite of the rotational
invariance of both magnetic and point interactions, their
superposition can give rise to anisotropies.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Relativistic two--particle one--dimensional scattering
problem for superpositions of delta potentials} \\ [5mm]
{\large T.~Alferova} \\ [2mm]
Physics Department, Gomel State University, \\
Sovetskaya Str. 102, 246699 Gomel, Belarus
\end{center}
\noindent
The covariant single-time equations of the quantum field theory are
formulated in the relativistic configurational representation. The
explicit forms of Green functions for scattering states are
computed in this representation. On the basis of derived
nonhomogeneous equations the scattering problem is exactly solved
for some potentials (combinations of zero-range potentials).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf On the spectral properties \\
of discrete Schr\"odinger operators} \\ [5mm]
{\large A. Boutet de Monvel} \\ [2mm]
Mathematique, Universit\'{e} Paris VII, 2, place Jussieu, F--75252
Paris
\end{center}
\noindent
We use the method of the conjugate operator to prove the limiting
absorption principle and the absence of the singular continuous
spectrum for the discrete Schr\"odinger operator. We also obtain
local decay estimates. Our results apply to a large class of
perturbating potentials $V$ tending arbitrarily slowly to zero at
infinity.
\vspace{8mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Lax--Phillips scattering theory for short \\ and long range
perturbations} \\ [5mm]
{\large Fernando Brambila Paz} \\ [2mm]
Department of Mathematics, Facultad de Ciencias, \\
UNAM, Mexico
\end{center}
\noindent
On 1966 Lax and Phillips present a geometric scattering theory for the
wave equation for perturbations with compact support and some others
conditions. There has been a lot of papers that drop some of the
conditions that Lax and Phillips assume on 1966. On 1982 Phillips
prove a Lax and Philips scattering theory for the wave equation for
short range perturbations of order $x^{-\beta}$, for $\beta>2$. We
will present a theory for short range perturbations and long range
perturbations that generalize the last article of Phillips. We will
use the Caffarelli inequality.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Inverse spectral theory \\ for self--adjoint extensions} \\
[5mm]
{\large J.~Brasche} \\ [2mm]
Institute of Applied mathematics, Bonn University, \\ Wegelerstr. 10,
D--52115 Bonn
\end{center}
\noindent Let $H$ be the Hamiltonian corresponding to a
zero--range interaction, e. g. a point interaction (the
interaction only takes place within a given discrete set
$\Gamma$) or a surface interaction (it only takes place
within some manifold $\Gamma$ with codimension 1). As it is
well known such an operator $H$ is not characterized via
some potential function $V$ but a self--adjoint extension
of the restriction of the free Hamiltonian on the space of
functions supported on the complement of $\Gamma$.
%
In inverse spectral theory (for regular Schr{\"o}dinger
operators) one investigates the question about how to
construct a potential $V$ such that the operator $-\Delta +
V$ has certain preassigned spectral properties. If one
studies the analogous problems for Hamiltonians $H$
describing a zero--range interaction then one is led to
the question about how to construct a self--adjoint
extension with preassigned spectral properties. I shall
discuss this question within a general operator theoretical
framework and shall also give explicit applications for
elliptic differential operators and, in particular,
Hamiltonians corresponding to zero--range interactions.
I shall present methods how to construct self--adjoint
extensions with preassigned eigenvalues, preassigned
absolutely continuous spectrum and for arbitrary
preassigned $\alpha \in [0,1]$ with non--empty
$\alpha$--dimensional spectrum.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Stark operators with some discontinuous interactions} \\
[5mm]
{\large Ph.~Briet} \\ [2mm]
Departement de math\'{e}matiques, Universit\'{e} de Toulon et du Var,
\\ F--83957 La Garde Cedex, France
\end{center}
\noindent
In this talk, we discuss the existence and the behavior of the
boundary value of the resolvent : $(H(F) -E +i0)^-1$, here $E$ is an
real energy and $H(F)$ is the following Stark operator $H(F)= - D^2 +
V + Fx$ defined on $L^2(\R)$. The potential $V$ is bounded and
contains some jumps periodically distributed on $\R$. This analysis
used recent results of Briet-Mourre which give a link between the
behavior of the resolvent when the energy reach the real line $\R$
and the asymptotic behavior of the generalized solutions of the
associated differential equation. In our model, we prove that these
solutions have asymptotically the same behavior as in the free
Stark problem. By using the criteria mentioned above, we prove the
existence of the boundary value of the resolvent and show that the
spectrum is purely absolutely continuous in this case.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf The Riccati equation and \\ exactly solvable Hamiltonians}
\\ [5mm]
{\large J.L.~Cari\~nena} \\ [2mm]
Depto. F\'{\i}sica Te\'orica, Univ. de Zaragoza, 50009 Zaragoza,
Spain
\end{center}
\noindent
By introducing the notion of $A$--related equations, $A$ being a
differential operator, we associate with the Schr\"odinger equation
a Riccati equation and study the solutions when the potential has
some specific properties.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Coherent states over $C^{M}$} \\
[5mm]
{\large G.~Chadzitaskos} \\ [2mm]
Department of Physics, Faculty of Engineering, \\
Czech Technical University, Prague
\end{center}
\noindent
We construct the coherent states over $\C^{M}$ for the case of
q-deformed har\-monic oscillator when $q^{M+1}\!=1$. Nilpotent
variables are then complex upper diagonal matrices. Corresponding
Berezin symbols and a star product are also constructed.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Exotic wavefunction aholonomy in one--dimensional quantum
mechanics of generalized pointlike potentials} \\ [5mm]
{\large T.~Cheon} \\ [2mm]
Laboratory of Physics, Kochi University of Technology,\\
Tosa Yamada, Kochi 782, Japan
\end{center}
\noindent
We show that
a potential consisting of three Dirac-delta functions
on the line with disappearing distances
can give rise to the discontinuity in
wave functions with the proper renormalization of
the delta function strength. This can be
used as a building block, along with the usual Dirac-delta,
to construct the most general three-parameter family of point
interactions which allow both discontinuity and asymmetry of
the wave function as the zero-size limit of local and
self-adjoint operator in one-dimensional quantum mechanics.
We then analyze the eigenvalue problem of a quantum particle on the line
with the generalized pointlike potential so constructed.
Several unusual characteristics of the system is unveiled.
In particular, it is shown that
the energy surface in the parameter space has a set
of singularities, around which different eigenstates are connected
in the form of paired spiral stairway.
An exemplar wave-function aholonomy is displayed where the ground
state is adiabatically turned into the second excited state
after cyclic rotation in the parameter space.
We discuss it's potential utility in the quantum dot experiments
and heterojunction of semiconductors.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf A rigorous proof of Gutzwiller's \\ trace formula} \\ [5mm]
{\large M.~Combescure} \\ [2mm]
Laboratoire de physique th\'eorique et des hautes energies, \\
Universit\'{e} Paris XI, Centre d'Orsay, Batiment 211, F--91405 Orsay
\end{center}
\noindent
For classically chaotic systems, the trace of suitable quantum
observables can be expanded semiclassically in powers of the Planck's
constant $\hbar\;$ (as $\hbar\to 0$). Under suitable assumptions,
these expansions involve the invariants in phase space of the
corresponding classical flow (energy surfaces with the corresponding
microcanonical measure, periodic trajectories $\dots$). We show how
the usual ``coherent states" of quantum mechanics and their
semiclassical propagation estimates allow to prove these expansions.
\vspace{20mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf One dimensional many--body problems \\ with contact
interactions} \\ [5mm]
{\large L.~Dabrowski} \\ [2mm]
SISSA, via Beirut 2--4, I--34014 Trieste, Italy
\end{center}
\noindent
One dimensional many--body problem with most general
two--particle contact interactions is considered.
Besides the pure (repulsive or attractive)
$\delta$-interaction two other one--parameter families
of integrable one dimensional many--body systems are
presented.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Singular continuous spectrum for substitution
Hamiltonians} \\ [5mm]
{\large D. Damanik} \\ [2mm]
FB Mathematik, J.--W. Goethe Universit\"at, D--60054 Frankfurt a.M.
\end{center}
\noindent
We consider discrete one-dimensional Schr\"odinger operators with
potentials generated by primitive substitutions. Purely singular
continuous spectrum with probability one is established provided that the
potentials have a local four-block structure.
\vspace{20mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Generalized Heisenberg--Dirac Hamiltonian} \\ [5mm]
{\large M.~Damnjanovi\'{c}} \\ [2mm]
Faculty of Physics, University of Beograd, Studentski trg 10--12 \\
P.O.Box 368, Beograd 11001, Yugoslavia
\end{center}
\noindent
The first order perturbative model of the spin independent
two--particle interaction term in the many--electron system for the
different orbital states of the electron is known as the
Heisenberg--Dirac Hamiltonian. Being the standard model for research
of phase transitions in statistical physics, it has attracted
attention recently in connection with spin ladders. Its
generalization to the one electron states being orbitally degenerated
is considered, and the spectrum is analyzed for the spin--ladder
system. The discussion on the separation of the bands is performed,
especially on the appearance of the spin--Peierls gap.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf On the semi-classical analysis of quantized ``chaotic"
maps on the torus} \\ [5mm]
{\large S.~De Bi\`evre} \\ [2mm]
Universite des Sciences et Technologies de Lille, \\
UFR de Mathematiques, 59655 Villeneuve d'Ascq, FRANCE
\end{center}
\noindent
The simplest hamiltonian dynamical systems that display chaotic behaviour
are various maps on the two-torus: toral automorphisms (``cat maps"), the Baker
transformation, sawtooth maps, and various kicked maps are examples. A short
review of the known results on the semi-classical spectral behaviour of
the quantized versions of these maps will be presented, as well as some recent
results on the non-commutativity of the $\hbar\to0$ and the $t\to\infty$
limits and the occurrence of various $\ln\hbar$ time-scales.
\vspace{20mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Generalized Pearson estimate \\ for wave operators } \\ [5mm]
{\large \underline{M.~Demuth}, S.~Eder} \\ [2mm]
Institut f\"ur Mathematik, TU Clausthal, D--38678
Clausthal--Zellerfeld
\end{center}
\noindent
Pearson's estimate is known for trace class differences of admissible
functions. This can be extended to sandwiched differences mapping
$L^\infty$ to $L^1$. The abstract theory can be used for differential
operators of order higher than two. That implies perturbation
estimates for the scattering operators also for such operators.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Solving the main equation of Poincar\'{e} invariant
quantum mechanics for a model potential}
\\ [5mm]
{\large V.V.~Andreev, \underline{O.M.~Deryuzhkova}} \\ [2mm]
Gomel State University, Belarus
\end{center}
\noindent
The model potentials of two-particle relativistic systems are
considered in the framework of Poincare-invariant quantum mechanics
(or relativistic Hamiltonian dynamics). A class of potentials is
obtained which reduce the main integro-differential equation of
Poincare-invariant quantum mechanics to the equation being an
analogue to equation of motion in radial field. Mass spectrum of
two-particle bound relativistic system is obtained for this class of
potentials.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Massive scalar field in an oscillating bounded region} \\
[5mm]
{\large J.~Dittrich} \\ [2mm]
Nuclear Physics Institute, Academy of Sciences, \v{R}e\v{z}, and \\
Doppler Institute, Czech Technical University, Prague
\end{center}
\noindent
Classical scalar field satisfying 1+1 dimensional Klein--Gordon
equation in a space interval of periodically oscillating length is
considered. Dirichlet boundary conditions at the endpoints are
assumed. The mass term is treated perturbatively.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%\documentstyle[12pt]{article}
%\textheight=24.7cm
%\textwidth=16.5cm
%\voffset=-2.5cm
%\hoffset=-1.4cm
%\renewcommand \baselinestretch{1.1}
%\begin{document}
%\def\bce{\begin{center}}
%\def\ece{\end{center}}
%\def\beq{\begin{eqnarray}}
%\def\eeq{\end{eqnarray}}
%\def\ben{\begin{enumerate}}
%\def\een{\end{enumerate}}
%\def\ul{\underline}
\def\ni{\noindent}
%\def\nn{\nonumber}
\def\bs{\bigskip}
%\def\ms{\medskip}
%\def\wt{\widetilde}
%\def\wh{\widehat}
%\def\brr{\begin{array}}
%\def\err{\end{array}}
%\def\dsp{\displaystyle}
%\pagestyle{empty}
\begin{center}
{\Large\bf The determinant anomaly in low--dimensional quantum
systems} \\ [5mm]
{\large E. Elizalde} \\ [2mm]
Consejo Superior de Investigaciones Cient\'{\i}ficas (CSIC),\\
IEEC, Edifici Nexus 201, Gran Capit\`a 2-4, 08034 Barcelona, Spain\\ and \
Dept. ECM and IFAE, Fac. de F\'{\i}sica,
Univ. de Barcelona, \\ Diagonal 647,
08028 Barcelona, Spain \\
{\it E-mail: elizalde@ieec.fcr.es}
\end{center}
\noindent
We have been working on possible physical applications (and implications)
of the non-commutative anomaly, which comes from the fact that the
determinant of the product of two differential operators is not equal
(in general) to the product of the individual determinants. We have
evaluated the partition function of a charged massive free Bose gas
at finite temperature within a path-integral approach. A new,
additional vacuum term has been found and its dependence on the mass
and chemical potential has been obtained. The presence of the new
term is seen to be crucial, in this example, for having the
factorization invariance of the regularized partition function.
The most simple case, however, involves a square root of a
Hamiltonian which is very close to the the harmonic oscillator.
Already for this one-space-dimensional operator, the noncommutative
anomaly is shown to be non-vanishing. A generalization to Dirac-like
operators and to harmonic oscillators is carried out. The anomaly is
shown to be always non-vanishing and can be explicitly expressed in
terms of Barnes zeta functions.
Results obtained in joint work with S. Zerbini, L. Vanzo, A. Filippi and
G. Cognola.
\bs
{\small
\ni 1. E. Elizalde, L. Vanzo and S. Zerbini,
Trento preprint UTF 394, hep-th/9701060,
to appear in {\em Commun. Math. Phys.} (1998).
\ni 2. E. Elizalde, A. Filippi, L. Vanzo and S. Zerbini,
hep-th/9710171 (1997).
\ni 3. E. Elizalde,
{\sl Ten physical applications of spectral zeta functions}, Lecture
Notes in Physics (Springer-Verlag, Berlin, 1995);
{\em J. Phys.} {\bf A30}, 2735 (1997). }
%\end{document}
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Linear Boltzmann equation as a kinetic limit of random
Schr\"odinger equation} \\ [5mm]
{\large L.~Erd\"os} \\ [2mm]
Courant Institute, NYU, 251 Mercer Str., New York, NY 10012
\end{center}
\noindent
We study a quantum particle in a random potential in two scaling
limits: the low density limit (or Boltzmann-Grad) and the weak
coupling (semiclassical) limit. The low density limit is the quantum
analogue of the Lorentz gas. In both cases, the phase space density
of the quantum evolution defined through the Wigner transform or the
Husimi function converges weakly to a linear Boltzmann equation for
all time with a collision kernel given by the quantum scattering
cross section. This is a joint work with H.-T. Yau.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Two results about vortices: ``smoke rings" in transport
and anomalous Pauli trapping} \\ [5mm]
{\large P.~Exner} \\ [2mm]
Nuclear Physics Institute, Academy of Sciences, \v{R}e\v{z}, and \\
Doppler Institute, Czech Technical University, Prague
\end{center}
\noindent
We present two loosely related results involving vortical
singularities in simple quantum mechanical systems; they have been
obtained in collaboration with P.~\v{S}eba, F.~Bentosela, and
V.~Zagrebnov. First we show that the probability current associated
with scattering solutions may exhibit vortical singularities. For a
localized interaction in three dimensions the singular lines form
closed loops; we illustrate this on examples involving point
interactions.
The other result concerns a Pauli electron interacting with a
magnetic field generated by a planar electric--current vortex. If the
effective gyromagnetic factor $\,g^*>2$, there is a bound state with
spin antiparallel to the field provided the current is strong enough.
Moreover, electron scattering on the vortex exhibits resonances which
get sharper as the current increases. On the other hand --- in
distinction to the localized flux--tube case --- there is no binding
for weak vortices.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf S--matrix statistics in quantum chaotic scattering: poles,
phaseshifts and time delays} \\ [5mm]
{\large Y. Fyodorov} \\ [2mm]
Fachbereich Physik, Universit\"{a}t--GH Essen, D--45117 Essen, Germany
\end{center}
\noindent
The scattering in quantum chaotic systems can be
characterized by an effective non-Hermitian random matrix
Hamiltonian. Complex eigenvalues of such a Hamiltonian are
S-matrix poles (resonances). We discuss recent progress in
studying statistics of these poles and their relation to
the statistics of related quantities such as phaseshifts,
Wigner time-delays and photodissociation cross-sections. \\
[3mm]
1. Y.V. Fyodorov and H.-J.Sommers, {\it J.Math.Phys.}
v.8 pp. 1918-1981 (1997) \\
2. Y.V.Fyodorov, M.Titov and H.-J.Sommers, Statistics of S-matrix
poles for chaotic systems with broken time reversal invariance: a
conjecture, {\it e-preprint: cond-mat/9802306} and in preparation \\
3. Y.V. Fyodorov and Y.Alhassid, Photodissociation in
Quantum Chao\-tic Systems: Random Matrix Theory of
Cross-Section Fluctuations, {\it e-preprint: cond-mat/9802105}
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Constructing formal solutions to commutator equations
by the method of resolvents and its applications in constructing
quasi--time operators} \\ [5mm]
{\large E.A.~Galapon} \\ [2mm]
National Institute of Physics, University of the Philippines
\\ Diliman, Quezon City, Philippines; {\em egalapon@nip.upd.edu.ph}
\end{center}
\noindent
A formalism based on the method of resolvents is developed in finding
a formal operator $O(q,p)$ in the Hilbert space spanned by the
position and momentum operators $q$ and $p$ satisfying the
equation $[f(q,p),O(q,p)]=g(q,p)$ for some given operators $f(q,p)$
and $g(q,p)$. Specifically explicit solutions are derived for
$f(q,p)=Q(q)$ and $f(q,p)=P(p)$ for some $g(q,p)$ in which $Q(z)$ and
$P(z)$ are analytic, entire functions of $z$. Furthermore the
formalism is extended in solving coupled commutator equations of the
form
$$
[Q(q),O(q,p)]=g_Q(q,p), \quad [P(q),O(q,p)]=g_P(q,p).
$$
Finally the formalism is applied in solving for the quasi--time
operator $T(q,p)$ conjugate to the Hamiltonian of the usual single
Hilbert space formulation of quantum mechanics, i.e.,
$[T(q,p),H(q,p)]=i$, in one--dimension. While $T(q,p)$ is well known
to be a non--Hermitian operator (which is the reason why we refer to
it as quasi--time operator in this work), it has recently been found
useful in constructing exact solutions to to operator differential
equations of the Heisenberg type (C.M.~Bender, {\em Contemporary
Math.} {\bf 160}, 31 (1994)); and in constructing operator invariants
used in a new perturbation scheme (H.~Lewis and W.~Lawrence, {\em
Phys. Rev.} {\bf A55}, 2615 (1997)). As an example the quasi--time
operator for the one--dimensional harmonic oscillator is explicitly
found and shown to be that found independently by Lewis and his
collaborators ({\em Phys. Rev. Lett.} {\bf 77}, 5157 (1996)), and by
Bender and Dunne ({\em Phys. Rev.} {\bf D40}, 2739 (1989)).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Dynamical localization and spectral properties.
Application to random and quasi--periodic Schroedinger operators } \\
[5mm]
{\large F.~Germinet} \\ [2mm]
Mathematique, Universit\'{e} Paris VII, 2, place Jussieu, F--75252
Paris
\end{center}
\noindent
We prove, for a large class of discrete as well as continuous random
Schr\"odin\-ger operators, and also for the quasi-periodic Mathieu operator,
a strong criterium of localization, called {\it dynamical localization},
meaning that, with probability one, for a suitable energy interval $I$
and for $q>0$,
$$
\sup_t r^q_{\psi,I}(t) \equiv \sup_t \langle P_I(H_\omega)\psi_t,
\ |X|^q P_I(H_\omega) \psi_t \rangle <\infty.
$$
Here $\psi$ is a function of compact support, $\psi_t=e^{-iH_\omega
t}\psi$ and $P_I(H_\omega)$ is the spectral projector of $H_\omega$
corresponding to the interval $I$. It is now known that Exponential
Localization of the Hamiltonian (pure point spectrum and exponential
localization of the eigenfuntions) is not sufficient to entail Dynamical
Localization (Del Rio, Jitomirskaya, Last, Simon). Therefore, one has to
ask for further spectral properties.
The result is obtained through the control of the localization length of
the eigenfunctions of the Hamiltonian, achieved by exploiting the fruits
of the same multi-scale analysis that usually yields the Exponential
Localization. It covers, in particular, the one dimensional discrete
Anderson model with a Bernoulli potential, and the Mathieu operator
$H_{\theta,\lambda,\omega} = -\Delta + \lambda
\cos(2\pi(\theta+x\omega))$, where $\lambda\geq 15$ and $\omega$ with
good Diophantine properties.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Band structure of the spectrum \\ in some models with a
magnetic field} \\ [5mm]
{\large V.~Geyler} \\ [2mm]
Department of Mathematics, Mordovian State University, \\ 430000
Saransk, Russia
\end{center}
\noindent
Explicitly solvable models of quantum-mechanical periodic systems
with a magnetic field are considered. We deal with two classes of
such models: 1) the point potential models; 2) the zero-range potential
models with internal structure. For the three-dimensional magnetic
Schrodinger operator with a periodic point potential the gap number
finiteness conjecture (Bethe - Sommerfeld conjecture) is justified
in the case of arbitrary rational flux and arbitrary finite number of
point scatterers in the Wigner - Seitz cell (under some technical
assumption about the period lattice). An evidence for a fractal
structure of the spectrum in the case of ``non-local" periodic point
perturbation is given (we stress that the corresponding Hamiltonian
is local in the ordinary sense). For the perturbation of a magnetic
Schrodinger operator by a chain of point scatterers (like a graphite
nanotube) we obtain a description of the spectrum and scattering states.
In addition, some results concerning periodic point perturbations in
the Lobachevsky space are discussed. In the framework of the models
with internal structure we study spectral properties of a periodic
array of mesoscopic rings with the Aharonov - Bohm vortices through
the rings. The spectral analysis shows some localization phenomena
in such arrays which are similar to the known localization phenomena
for the chaotic orbits in the periodic arrays of quantum antidots.
Some numerical results for the above-mentioned models concerning
calculation of the ``flux - energy" diagrams of the Hofstadter butterfly
type are presented.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Quantum chaos and $SU(3)$ dynamics} \\ [5mm]
{\large \underline{S.~Gnutzmann}, M.~Kus} \\ [2mm]
Fachbereich Physik, Universit\"at GH Essen, 45117 Essen, Germany
\end{center}
\noindent
We describe the classical limit of quantum mechanical systems defined
on irreducible representations of $SU(3)$. Such a limit leads to a
phase space of dimension four or six depending on the irreducible
representations used in the limit. The same quantum mechanical
Hamiltonian may lead to a classical dynamics which is integrable on
the four-dimensional phase space but chaotic in the six-dimensional
case. Evidence for this behavior is found in the level spacing
distributions of the quantum mechanical Hamiltonian which is Poisson
distributed in representations belonging to a four-dimensional
classical limit but shows level repulsion in representations
belonging to a six-dimensional classical limit.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Quantum kinematics of vortex filaments} \\ [5mm]
{\large G.A.~Goldin} \\ [2mm]
Departments of Mathematics and Physics, Rutgers University
\end{center}
\noindent
The Poisson structure for extended vortex loops and arcs in
an ideal, incompressible two-dimensional superfluid is analyzed
in detail. Canonical coordinates and momenta are found on the
coadjoint orbits of the group of area-preserving diffeomorphisms
of the plane (symplectomorphisms) associated with such
extended configurations of vorticity. The quantized theory is
then constructed, in the simplest case of ``bosonic'' vortex loops.
We introduce a Fock-like structure, with nonlocal creation
and annihilation field operators that satisfy appropriate commutator
brackets with each other and with the local current densities.
A discrete approximation permits mappings from the infinite-dimensional
coadjoint orbit description to related finite-dimensional theories.
This talk includes results reached jointly with Robert Owczarek and
David H. Sharp, Los Alamos National Laboratory.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf KAM theorem for nonlinear \\ [1mm] Schr\"odinger operator}
\\ [5mm]
{\large B.~Grebert} \\ [2mm]
Departement de Math\'ematiques, Universit\'{e} de Nantes, \\
2, rue de la Houssiniere, F--44322 Nantes
\end{center}
\noindent
We prove the persistence of finite dimensional invariant tori
associated with the defocusing nonlinear Schr\"odinger equation under
small Hamiltonian perturbations. The invariant tori are not
necessarily small. To do this, we first prove the existence of global
Birkhoff variables and then apply a refined version of a KAM Theorem
in infinite dimension of S. Kuksin.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Instability of relativistic matter \\ with self--generated
magnetic field} \\ [5mm]
{\large M.~Griessemer} \\ [2mm]
Mathematik, Universit\"at Regensburg, D--93040 Regensburg \\
{\em Marcel.Griesemer@mathematik.uni-regensburg.de}
\end{center}
\noindent
The Breit potential is often employed in models of relativistic
matter to account approximatively for the interaction of the
electrons with the self-generated magnetic field. For a model of
stable relativistic matter, where electrons are vectors in the
positive spectral subspace of the free Dirac operator, it is shown
that both, the inclusion of the Breit potential as well as the
inclusion of the quantized radiation field leads to instability (of
the second kind) for all values of the fine-structure constant.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Spectral nature of the Schr\"odinger operator with
periodic magnetic field (rational flux)} \\ [5mm]
{\large M.~Gruber} \\ [2mm]
Institute of Mathematics, Humboldt University at Berlin, \\
Unter den Linden 6, D-10099 Berlin
\end{center}
\noindent
The Schr\"odinger operator on $\R^n$ with periodic electric
potential is well known to have absolutely continuous spectrum. For the
Schr\"odinger operator with periodic magnetic field (zero flux), absence
of singular continuous spectrum was shown by Hempel and Herbst using
Bloch theory for function spaces. Recently Birman and Suslina ($n=2$)
and Sobolev ($n=3$) showed absence of eigenvalues in this case.
For rational flux eigenvalues may exist. Employing Bloch theory for
spaces of sections of vector bundles we show absence of singular
continuous spectrum in this case.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Time--dependent scattering \\ on fractal measures} \\ [5mm]
{\large Ch.-A.~Guerin} \\ [2mm]
Centre de Physique Th\'eorique, CNRS, Marseille Luminy
\end{center}
\noindent
In this paper we study the time-evolution for the
Schr{\"o}dinger equation and the wave equation on the line when the
interaction term is a fractal measure. First, we extend the usual
one-dimensional potential scattering formalism to interactions defined as
measures. Then we show how to retrieve
information on the fractality of the interaction term from
time-dependent scattering data. In the case of the Schr{\" o}dinger
equation we shall obtain the wavelet correlation dimension of the
scatterer. For the wave equation the whole set of generalized
multifractal dimensions can be recovered, provided the scatterer
actually is fractal (=non smooth). In this latter case, we also show
how the reflected wave packets can be interpreted in term of wavelet
transform of the interaction.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Inverse scattering at fixed energy \\ for layered media} \\ [5mm]
{\large J.-C.~Guillot} \\ [2mm]
D\'epartement de Math\'ematique, LAGA, Institut Galil\'ee, \\
Universit\'e Paris-Nord, Avenue J.-B. Cl\'ement, F-93430 Villetaneuse
\\ [3mm]
{\em no abstract submitted}
\vspace{2mm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf The classical limit \\ of quantum dissipative systems} \\
[5mm]
{\large Z.~Haba} \\ [2mm]
Institute of Theoretical Physics, Wroc{\l}aw University, Poland
\end{center}
\noindent
We discuss the Lindblad equation for the density matrix where the
dissipation is linear in the position operator. We consider a
potential which is a bounded perturbation of the harmonic oscillator.
We show that the perturbation of the potential leads to an analytic
perturbation of the Wigner distribution. Then the Wigner distribution
of the quantum dissipative system tends (uniformly in time) to the
classical phase space distribution of the classical dissipative
system (if the initial distribution converges when $\hbar\to 0$).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Scattering calculus and the $N$--body problem} \\ [5mm]
{\large A.~Hassell} \\ [2mm]
Institute for Mathematics and its Applications, \\ Australian National
University, Canberra ACT 0200
\end{center}
\noindent
In this talk I will introduce the scattering calculus --- that is,
a class of pseudodifferential operators --- on Euclidean space, and
outline how it can be used to solve various problems in N body
scattering theory. I will discuss the existence and asymptotics of
plane wave-type eigenfunctions and the structure of the scattering
matrix.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf On spectral properties \\ of Harper--like models} \\ [5mm]
{\large D.J.L.~Herrmann} \\ [2mm]
Theoretische Fysica, KU Nijmegen, Postbus 9010 \\
NL--6500 GL Nijmegen, The Netherlands
\end{center}
\noindent
We study spectral properties of Harper--like models by algebraic and
combinatorial methods and derive sufficient conditions for the
existence of spectral gaps with qualitative estimates. For this class
the Chambers relation holds and we obtain an analytic expression for
the representation depending part. The models corresponding to the
rectangular and triangular lattice are studied. For the second one we
show that one class of spectral gaps are open for magnetic fields
with ``rational magnetic flux per unit cell'' and a quantitative
estimate for the gap widths is given. Further, for ``irrational
magnetic flux'' fulfilling some Liouville condition the spectrum is a
Cantor set.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf An example of a second--kind phase transition for some
discrete Schr\"odinger operators with magnetic potential} \\ [5mm]
{\large Y.~Higuchi, T.~Shirai, \underline{H.~Hirata}} \\ [2mm]
Department of Mathematics, Sophia University,\\
7-1, Kioicho, Chiyoda-ku, Tokyo 102-8554, JAPAN; {\em
h-hirata@mm.sophia.ac.jp}
\end{center}
\noindent
We consider the discrete Schr\"odinger operators $ H_{\theta}$ on a
graph with a magnetic potential defined as follows:
$$
H_{\theta}f(x)=\frac{1}{m(x)}\sum_{y\in N_{G}(x)}
\bigl(\exp(i\theta([x,y])f(y) -f(x)\bigr).
$$
We observed the spectral bottoms of these operators and found an
example which leads to an interesting transition with the variation
of the magnetic potential.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Correlated Wegner inequalities and localization for
long--range and correlated potentials} \\ [5mm]
{\large J.-M.~Combes, \underline{P.D.~Hislop}, E.~Mourre} \\ [2mm]
Centre de Physique Th\'eorique, C.N.R.S. \\
Luminy Case 907, 13288 Marseille FRANCE
\end{center}
\noindent
We use a differential inequality method to prove correlated Wegner
estimates for local Hamiltonians with Anderson-type potentials
constructed with either long-range, single-site potentials and/or
correlated coupling constants. These are estimates on the joint
probability distributions of correlated random variables constructed
from finite-volume Hamiltonians. The results are applied to prove
exponential localization for random Hamiltonians, with Anderson-type
potentials and correlated coupling constants, in energy intervals
near the band edges of the spectrum of the unperturbed Hamiltonians.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Convergence and localization properties of the spectral
expansion of Schr\"odinger and Dirac operators} \\ [5mm]
{\large M.~Horv\'ath} \\ [2mm]
Mathematical Institute, Technical University of Budapest, \\
Stoczek ucza 2, H-1111 Budapest, Hungary
\end{center}
We study spectral expansions
$$
f=\sum_nc_n\Psi_n\qquad,\qquad c_n=\int\Psi_n^*f\,dx;
\eqno{(1)}
$$
where the $\Psi_n$ are eigenfunctions of the Hamiltonian:
$
H\Psi_n=\lambda_n\Psi_n.
$
Such orthogonal expansions appear as the solution of the
Schr\"odinger resp. Dirac equation; the $\Psi_n$ represent stationary
states with energy level $\lambda_n$.
We provide two results. In case of stationary Dirac operator with
smooth potential we show the local uniform convergence of the
expansion (1) if the function expanded is sufficiently smooth. Also
we provide a version of the Riemann localization principle,
well-known for trigonometric Fourier series.\newline The singularity
of the potential causes many difficulties in spectral theory of
quantum mechanical operators. We investigate here the stationary
Schr\"odinger operator with singular potential. We give sharp
conditions, depending on the type of the singularity, under which the
expansions (1) converge in norm of smooth function spaces.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Representation of quantum mechanical resonances in the
framework of Lax--Phillips scattering theory} \\ [5mm]
{\large \underline{L.P.~Horwitz}, Y.~Strauss} \\ [2mm]
School of Physics, Raymond and Beverly Sackler Faculty \\ of Natural
Sciences Tel Aviv University, Ramat Aviv, and \\
Department of Physics, Bar Ilan University, Ramat Gan, Israel
\end{center}
\noindent
We discuss the quantum Lax--Phillips theory of scattering and unstable
systems. In this framework, the decay of an unstable system is
described by a semigroup. The spectrum of the generator of the
semigroup corresponds to singularities of the Lax--Phillips
$S$--matrix. In the case of discrete (complex) spectrum of the
generator of the semigroup, associated with resonances, the decay law
is exponential. The states corresponding to these resonances
(eigenfunctions of the generator of the semigroup) lie in the
Lax--Phillips Hilbert space, and therefore all physical properties of
the resonant states can be computed. We show, for pointwise (in $t$)
models, that the Lax--Phillips $S$--matrix is unitarily related to the
$S$--matrix of standard scattering theory by a family of unitary
transformations parametrized by the spectral variable $\sigma$ of the
Lax--Phillips theory. Analytic continuation in $\sigma$ has some of
the properties of a method developed some time ago for application to
dilation analytic potentials.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf An upper bound on the density of states for continuum
Schr\"odinger operators with Gaussian random potentials and magnetic
fields} \\ [5mm]
{\large \underline{T.~Hupfer}, H.~Leschke, P.~M\"uller, S.~Warzel} \\
[2mm]
Institut f\"ur Theoretische Physik, Universit\"at
Erlangen--N\"urenberg, \\ Staudtstr.~7, D--91058 Erlangen, Germany
\end{center}
\noindent
The Wegner estimate derived earlier [1] for multidimensional continuum
Schr\"odinger operators with Gaussian random potentials is extended
to the case of magnetic fields. It allows to construct an explicit
upper bound on the density of states which is discussed in some
detail. Specializing to two space dimensions and a constant magnetic
field, we also prove an upper bound on the density of states
restricted to the lowest Landau level. \\ [3mm]
[1] \quad W.~Fisher, T.~Hupfer, H.~Leschke, P.~M\"uller: Existence of
the density of states for multi--dimensional continuum Schr\"odinger
operators with Gaussian random potentials, {\em Commun.Math.Phys.}
{\bf 190} (1997), 133--141.
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Quantum Heisenberg chain \\ with elliptic exchange} \\ [5mm]
{\large V.I.~Inozemtsev} \\ [2mm]
Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
\end{center}
\noindent
It is shown that for a one-parameter set ${\cal H}_{N}$ of linear
combinations of $N(N-1)/2$ elementary transpositions $\{P_{jk}\}$
$(1\leq j 0$ be a self-adjoint operator in a separable complex
Hilbert space ${\cal H}$ and let ${\cal H}_{-k} \supset {\cal H}_0
\equiv {\cal H} \supset {\cal H}_k, \ k \geq 0$ be $H_0$-scale of
spaces. A symmetric operator $T$ acting from ${\cal H}_k$ to ${\cal
H}_{-k}, \ k>0$ gives a singular perturbation of $H_0 $ if there
are vectors $\varphi \in DomT \cap DomH_0$ such that $T\varphi \in
{\cal H}_{-k} \setminus {\cal H}.$ By definition a self-adjoint
singular perturbation $T \in {\cal H}_{-1}$-class if the subspace
$(RanT)^{cl,-1}$ \ ($cl, -k$ means the closure in $ {\cal H}_{-k}$)
has zero intersection with ${\cal H}$, and $T \in {\cal
H}_{-2}$-class if $(RanT)^{cl,-2} \cap {\cal H}_{-1} = \{ 0 \}.$
Let $T \in {\cal H}_{-2}$-class and $H_T$ is defined by the Krein's
formula. Let operators $T_n \in {\cal H}_{-1}$-class, \ $n=1,2,...$
and $H_n:= H_0 \dot + T_n$ is defined in the sense of quadratic
forms. Assume $T_n \to T$ in the strong resolvent sense. Then $H_n
\to H_0 $, as $n \to \infty$. We find a regularization $T_{n,r}$
such that the sequence $H_0 \dot + T_{n,r}$ goes to $ H_T$ in the
strong resolvent sense. Besides we find conditions for $T$ when $H_T$
has an additional point spectrum.
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Explicit solution to some
second--order differential and $q$--difference eigenvalue equations
related to $sl_2$ and $U_q(sl_2)$} \\ [5mm]
{\large I.~Krasovsky} \\ [2mm]
Max Planck Institute for Complex Systems, \\ N\"othnitzerstr. 38,
D--01187 Dresden
\end{center}
\noindent
We find a second--order differential and a $q$--difference operator
whose full set of eigenvalues and eigenvectors in the space of
polynomials of degrees at most $N$ can be written explicitly. The
eigenvectors are the generating functions of the dual Hahn (for the
differential operator) and dual $q$--Hahn (for the $q$--difference
operator) orthogonal polynomials. The differential and
$q$--difference operators are expressed in terms of the generators of
$sl_2$ and $U_q(sl_2)$ algebras, respectively, represented in the
space of polynomials.
The corresponding differential eigenvalue equation is equivalent to a
quasi--exactly solvable one--dimensional Schr\"odinger equation with
explicit solutions. Connection between the $q$--difference eigenvalue
equation and the Azbel--Hofstadter problem is indicated.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Solvable models in non--Abelian quantum kinematics and
dynamics} \\ [5mm]
{\large J.~Krause} \\ [2mm]
Pontificia Universidad Cat\'olica de Chile, {\em e-mail:
jkrause@chopin.fis.puc.cl}
\end{center}
\noindent
Plausible kinematic foundations of quantum dynamics are discussed in a
heuristic manner, in which the quantum rule stems directly from the
configuration symmetries of a system. Upon quantizing the complete
symmetry group of a system, irreducible generalized configuration-state
representations can be calculated, whose transition amplitudes yield the
propagation kernel in the configuration spacetime of the system. Thus,
solvable models result from a set of generalized Schr\"odinger equations
corresponding to the superselection rules dictated by the respective
quantized groups, in which the propagation kernels are obtained as invariant
integrals over the group manifold. No path integrals are needed in this
approach, since sums over physical world-lines are evaluated, instead of
sums over arbitrary paths, for obtaining the propagation kernel of systems
having a classical Lagrangian analog. The heuristic argument strongly
suggests that non-Abelian quantum kinematics is able to produce reasonable
solvable models, at least for all linear Newtonian systems. The attained
quantum-kinematic formalism, however, is completely general and does
not depend on this particular interpretation, for it seems to contain the
formalism of standard quantum mechanics as a very special case. \\ [3mm]
J. Krause, {\sl Phys. Rev. A \bf 54}, 4691 (1996). \\
J. Krause, {\sl Int. J. Theor. Phys. \bf 36}, 847 (1997). \\
J. Krause, {\sl Int. J. Theor. Phys. \bf 37}, (February issue, 1998).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Birman-Schwinger analysis for bound states in a pair of
parallel quantum waveguides with a semitransparent boundary} \\ [5mm]
{\large D.~Krej\v{c}i\v{r}\'{\i}k} \\ [2mm]
Faculty of Mathematics and Physics, Charles University, \\
V Hole\v{s}ovi\v{c}k\'ach 2, 18000 Prague
\end{center}
\noindent
We study a double quantum waveguide consisting of two parallel ducts
separated by a $\delta$ barrier with the ``coupling constant" which
may vary longitudinally. We construct the Birman--Schwinger theory
for the case when the latter has the same limit in both directions,
and use it to discuss properties of the discrete spectrum.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf On spectral problems \\of photonic crystals theory} \\ [5mm]
{\large P.~Kuchment} \\ [2mm]
Mathematics Department, Wichita State University, Wichita, KS
67260--0033, U.S.A.
\end{center}
\noindent
A photonic crystal is an artificial periodic dielectric medium whose
main feature is existence of a gap in the frequency spectrum of
electromagnetic waves. Creation of photonic crystals promises to
bring about revolutionary changes in technology. The task of creating
photonic crystals leads to interesting unexplored problems of
spectral theory of scalar and vector elliptic differential operators
with periodic coefficients and of differential and
pseudo-differential operators on periodic graphs. Similar problems
also arise in mesoscopic physics (``quantum wires"). The talk
contains a survey of recent results and approaches in this area.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf A microwave realization \\ [1mm] of the Hofstadter
butterfly} \\ [5mm]
{\large U.~Kuhl} \\ [2mm]
FB Physik der Universit\"at Marburg, Renthof 5, D--35032 Marburg
\end{center}
\noindent
The transmission of microwaves through an array of 100 scatterers
inserted into a waveguide is studied. The length of each scatterer
could be varied individually thus allowing the realization of
arbitrary scattering arrangements. For periodic sequences with
varying period length the found transmission bands reproduced the
Hofstadter butterfly, originally predicted for the spectra of
conduction electrons in strong magnetic fields. In the experiment it
was used that the same transfer matrix formalism is applicable both
to the microwave and the electronic system.
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Few-body Krein's formula} \\ [5mm]
{\large P.~Kurasov} \\ [2mm]
Dept. of Mathematics, Stockholm University, 10691 Stockholm, SWEDEN
\end{center}
\noindent
Standard Krein's formula relates the resolvents of two
selfadjoint extensions of one symmetric operator.
This formula can be used to calculate the resolvents
of singularly perturbed operators and of the generalized
extensions of symmetric operators. The operators
with the singular interactions concentrated on the
sets of zero measure can be defined first restricting
the original operator to a certain symmetric one and
then extending the symmetric operator to another
selfadjoint operator.
We concentrate our attention to the operators
describing the few-body quantum mechanical
problems. The corresponding
symmetric operators
have infinite deficiency indices and possess special translational
properties.
It is shown how to construct the selfadjoint
extensions and generalized selfadjoint extensions
possessing the same translational properties, i.e.
how to separate the Hamiltonians describing the few-body
problems from the set of all selfadjoint extensions
of the symmetric operators. Concrete examples of such
operators describing three particle systems are discussed. \\ [3mm]
1. P.Kurasov, Energy dependent boundary conditions and the
few-body scattering problem, {\it Rev. Math. Phys.}
{\bf 9} (1997), 853-906.
\\
2. S.Albeverio and P.Kurasov, Solvable Shr\"odinger
type operators: singular perturbations of differential
operators, in preparation.
\\
3. P.Kurasov and B.Pavlov, Few-body Krein's formula,
in preparation.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf A classicist's opinion on quantum chaos} \\ [5mm]
{\large P.~Leach} \\ [2mm]
Department of Mathematics and Applied mathematics, \\
University of Natal, Durban 4041, South Africa
\end{center}
\def\pa{{\partial\over\partial a}}
\def\pt{{\partial\over\partial t}}
\def\pu{{\partial\over\partial u}}
\def\bq{\begin{eqnarray}}
\def\eq{\end{eqnarray}}
\def\n{\nonumber}
\def\z{&=&}
\def\ha{\mbox{$\frac{1}{2}$}}
\def\gW{\Omega}
\def\({\left (}
\def\){\right )}
\def\oqr{\mbox{$\frac{1}{4}$}}
\def\f{\frac}
\noindent
A classical Hamiltonian, $H$, possesses a Noether symmetry
%\[
$G = \tau\pt+\eta_i\pa{}{q_1}$
%\]
if there exists a function, $f$, such that
\[
\dot{f}=-\dot{\tau}H-\tau\pa{H}{t}-\eta_i\pa{H}{q_i}+\dot{\eta_i}p_i.
\]
Corresponding to $G$ there exists the Noetherian integral
%\[
$I=f+\tau H-\eta_ip_i$.
%\]
From Liouville's theorem the Hamiltonian of an $n$-degree of freedom
system is integrable if there exist $n$ independent first integrals.
An integrable system displays regular behaviour in its phasespace
trajectories.
The time-dependent Schr\"odinger equation is
\[
Hu=i\pa{u}{t}.
\]
The Lie point symmetries of the time-dependent Schr\"odinger equation
have the form
\[
G = \tau(t,q,u)\pt+\eta_i(t,q,u)\pa{}{q_1}+\zeta(t,q,u)\pu
\]
and are
\bq
G_1 \z u\pu\qquad\mbox{(homogeneity)}\n\\
G_2 \z f(t,q)\pu\qquad\mbox{(solution)}\n\\
G_i,&& i=1,k\leq \ha\(n^2+3n+6\)\qquad\mbox{($n$ dim system)}\n
\eq
and these last are related to the Noether symmetries of the classical
Hamiltonian according to
\bq
G_N \z a(t)\pt + \(\ha\dot{a}{\bf q} + {\bf b}\times{\bf q} +
c(t)\).\nabla \n\\
G_L \z G_N + iu\(\oqr\ddot{a}r^2+\dot{\bf c}.{\bf q}+d(t)+\ha
i\dot{a}\)\pu.\n
\eq
The time-dependent wave functions for the Schr\"odinger equation of
the form
\[
H = \ha\({\bf p}^2 + \gW^2r^2+\f{h^2}{r^2}\)
\]
are generated by the Noether-related Lie symmetries.
The absence of such symmetries indicates classical chaos.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Exactly solvable quantum mechanical potential
problems} \\ [5mm]
{\large G.~L\'evai} \\ [2mm]
Institute of Nuclear Research of the Hungarian Academy
of Sciences, \\ Pf. 51, H-4001 Debrecen, Hungary
\end{center}
\noindent
We discuss a method of generalizing relatively simple exactly
solvable potentials to ones with more flexible potential shapes
and illustrate it with the example of the generalized Coulomb
potential, which contains both the Coulomb and the harmonic
oscillator potentials (in $D$ dimensions) as special limiting cases.
We show that a number of results obtained for this potential
reduce to well-known formulae in the two
limits. Examples for this include the $S$-matrix,
a three-term recurrence relation for the matrix elements of the
Green's operator and a generalized Coulomb--Sturmian basis
associated with an $su(1,1)$ algebra. The Coulomb--oscillator
connection can also be
interpreted in a straightforward way in terms of this construction.
This potential offers an ideal testing ground for the singularities
at the origin. When formulated for $D=1$, it can be used to study
the peculiarities of the
one-dimensional Coulomb potential $V(x)=-c \vert x\vert^{-1}$,
which it contains as a special case.
We briefly mention further fully analytical results for
even wider classes of potentials obtained using the techniques of
supersymmetric quantum mechanics.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Adiabatic curvature, chaos, and \\ the deformations of
Riemann surfaces} \\
[5mm]
{\large P.P.~L\'evay} \\ [2mm]
Department of Theoretical Physics, Institute of Physics, \\
Technical University of Budapest, H--1521 Budapest, Hungary
\end{center}
\noindent
Parametrized families of Landau Hamiltonians on leaky tori
${\Sigma}_{(g,n)}$
(i.e Riemann surfaces with genus $g\geq 1$ and number of cusps $n$)
corresponding to classically chaotic families of geodesic motion in
constant magnetic field
are investigated.
The parameters
describe deformations of these surfaces.
It is shown that the
adiabatic curvature two-form
of the lowest Landau level
is the sum of two (three) terms in the compact ${\Sigma}_{(g,0)}$
(noncompact
${\Sigma}_{(g,n)}, n\neq 0$)
case .
The first term is proportional to the natural symplectic form
on deformation space multiplied by
a quadratic expression of the magnetic field.
For both the compact and non-compact cases we have a fluctuating
term reflecting the chaotic nature of the geodesic motion, except
for $g=1$ $n=0$ (torus) where have no fluctuating term
the geodesic motion being integrable in this case.
For the non-compact case we have an extra third term modifying the
symplectic
structure
whose physical meaning is not clear.
The possibility of interpreting our result
as the curvature analogue of the well-known trace formulas is
emphasized.
As an illustration the $g=1$, $n=1$ case (Gutzwiller's leaky torus) is
worked out in some detail.
Connections with the viscosity properties of quantum Hall fluids on
such surfaces
is also pointed out.
An interesting possibility in this respect is the fractional
quantization of certain
components of the viscosity tensor of such fluids.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Resonances in a box}
\\ [5mm]
{\large B. Meller} \\ [2mm]
Facultad de F\'{\i}sica, P.U. Cat\'olica de Chile
\end{center}
\noindent
Consider a Schr\"odinger Operator on the line and then restrict it to
a box by imposing Dirichlet boundary conditions at $\{\pm\ell\}$. In
the numerical study of the dependence of the eigenvalues on $\ell$,
for large $\ell$, at certain energies pronounced ``plateaus''
interspersed with gaps are observed. These energy levels are expected
to be resonant energies for the operator on the whole line. In the
case of shape resonances this is proven in the semi-classical limit.
Bounds on the gaps are also given.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Schr\"odinger operators with local point interactions
in dimension one} \\ [5mm]
{\large V.~Mikhailets} \\ [2mm]
Institute of Mathematics, Warsaw University at Bia{\l}ystok, \\
ul. Akademicka 2, PL--15267 Bia{\l}ystok, Poland
\end{center}
\noindent
The talk is devoted to the Schr\"odinger operators on the real line
with local point interactions on an infinite discrete set. The problem
of the self-adjointness for these operators is investigated in a general
case.
Localization and structure of the spectrum are studied by
operator-theo\-retical methods. In particular some unexpected
connections between negative parts of spectra of the Schr\"odinger
operator
$$
H=-\frac{d^2}{dx^2}+\sum_{n=-\infty}^{\infty}a_{n}\delta(x-x_{n}),
$$
with $0 < d \leq x_{n+1}-x_{n}\leq d^{-1},\; a_n\in
%{\mathbb R}
\R, \; n \in
%{\mathbb Z}
\Z $ in the space $L^2(
%{\mathbb R}
\R)$ and the infinite Jacobi matrix $J$ in
the Hilbert space $l^2(
%{\mathbb Z}
\Z)$ with non-zero elements
$$
\left\lbrace-\,\frac{1}{x_{n}-x_{n-1}},\;a_{n}+\frac{1}{x_{n}-x_{n-1}}+
\frac{1}{x_{n+1}-x_{n}},\; -\,\frac{1}{x_{n+1}-x_{n}}\right\rbrace
$$
in the $n$-th row are found. The absolutely continuous of the
periodic Schr\"o\-dinger operators is proved in the most general case.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Symmetry classification of carbon nanotubes} \\ [5mm]
{\large \underline{I.~Milo\v{s}evi\'{c}}, R.~Sredanovi\'{c},
M.~Damnjanovi\'{c}} \\ [2mm]
Faculty of Physics, University of Beograd, Studentski trg 10--12 \\
P.O.Box 368, Beograd 11001, Yugoslavia
\end{center}
\noindent
Quasi--onedimensional members of the fullerene family, nanotubes
(based on the best studied molecule $C_{60}$) are classified
according to the line group symmetry and their vibrational spectra
are calculated. Their symmetries, described by the line groups, have
farreaching consequences on the form of the possible Hamiltonians of
such systems, as well as on the spectra and eigenvalues of the
Hamiltonians. Generalization of the Bloch's theorem has been derived
and used to simplify this problem. Also, some other considerations
enable to simplify the calculation as much as to reduce it in a way
that numerical calculations become radically simplified. The tubes'
symmetries, described by the helicoidal line groups, are extensively
used: numerical calculations have been performed by $POLSym$, the
software developed for calculation of physical quantities of polymers
(i.e. for research on systems having the line group symmetry). On the
basis of the semiempirical potentials further exact quantum
mechanical calculations of mechanical, electronic and optical
properties have been carried out. Since the modified group projector
technique has been used there has been no need for ``the (next)
nearest neighbour" based approximations: all orders of the interatom
interactions have been exactly dealt with.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf ``Gapology" for photonic crystals} \\ [5mm]
{\large A.~Moroz} \\ [2mm]
FOM Institute for Atomic and Molecular Physics, \\
Kruijslaan 407, NL--1098 Amsterdam, The Netherlands
\end{center}
\noindent
Using the photonic analogue of the Korringa--Kohn--Rostocker %(KKR)
method, the spectrum of electromagnetic waves in a three--dimensional
fcc lattice of dielectric spheres with various dielectric contrast is
analyzed. We confirm the main features of the spectrum obtained by
the plane wave method, namely that for a sufficiently high dielectric
contrast a full gap opens in the spectrum between the eights and
ninth bands if the dielectric constant $\varepsilon_s$ of spheres is
lower than the dielectric constant $\varepsilon_b$ of the background
medium. If $\varepsilon_s> \varepsilon_b$, no gap is found in the
spectrum. However, in contrast to the plane--wave method, the gap
width is completely determined at the W point where are the upper and
lower boundaries of the gap. The lowest dielectric contrast
$\varepsilon_b/\varepsilon_s$ for which a full gap opens in the
spectrum is $7.487\pm 0.0004$. An empirical formula which describes
the gap width as a function of the dielectric contrast and the
filling fraction is established.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Operator interpretation of resonances arising in spectral
problem for $2\times 2$ matrix Hamiltonians} \\ [5mm]
{\large A.~Motovilov} \\ [2mm]
Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
\end{center}
\noindent
We study resonances generated
by Hamiltonians of the $2\times2$ matrix form
$$
%
{\bf H}=\left(\begin{array}{lr}
%
A_0 & B_{01} \\
B_{10} & A_{1}
%
\end{array}\right)
%
$$
It is assumed that the operator $\bf H$ acts in an orthogonal
sum ${\cal H}={\cal H}_0\oplus{\cal H}_1$ of the channel
Hilbert spaces ${\cal H}_0$ and ${\cal H}_1$ while the entries
$A_0:{\cal H}_0\rightarrow{\cal H}_0$, and $A_1:{\cal
H}_1\rightarrow{\cal H}_1$, are self-adjoint operators. The
couplings $B_{ij}:{\cal H}_j\rightarrow{\cal H}_i$, $i{\neq}j$,
$B_{01}=B^*_{10}$, are assumed to be bounded operators. We are
especially interested in the physically typical case where the
spectrum of, say, $A_1$ is partly or totally embedded into the
absolutely continuous spectrum of $A_0$ and the transfer
function $M_1(z)=A_1-z+V_1(z)$, where
$V_1(z)=B_{10}(z-A_0)^{-1}B_{01}$, admits analytic continuation
(as an operator-valued function) through the cuts along branches
of the absolutely continuous spectrum of the entry $A_0$ into
the unphysical sheet(s) of the energy $z$ plane.
We construct an operator-valued function $V(X)$
on the space of operators in ${\cal H}_1$ possessing
the property: $V(X)\psi_1=V(z)\psi_1$ for any eigenvector
$\psi_1$ of $X$ corresponding to an eigenvalue $z$ and then
prove solvability of the equation
$$
H_1=A_1+V(H_1).
$$
Using the fact that the root vectors of the solutions $H_1$ of
this equation are at the same time such vectors for $M_1(z)$, we
prove completeness and even basis properties for the root
vectors (including those for the resonances). The results
obtained allow immediate applications in particular to the
scattering problems for multichannel Schr\"odinger equation.
The report is based on results of joint work with
Prof. R. Mennicken~[1]. \\ [3mm]
{\small\noindent
1. R. Mennicken, A. K. Motovilov, {\em Operator interpretation of
resonances arising in spectral problems for ${2}\times{2}$
operator matrices}, LANL E-print \mbox{\tt
funct-an/9708001}.}
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Aspects of a localization proof for continuum
Schr\"odinger operators with Gaussian random potentials}
\\ [5mm]
{\large W.~Fischer, H.~Leschke, \underline{P.~M\"uller}} \\ [2mm]
Institut f\"ur Theoretische Physik, Georg-August-Universit\"at, \\
D--37073 G\"ottingen, Germany
\end{center}
\noindent
Extending ideas presented earlier [1,2] we have proved that
multidimensional continuum Schr\"odinger operators with certain
Gaussian random potentials have almost surely pure-point spectrum at
sufficiently low energies. As stressed in [1], the proof relies
essentially on controlling correlations of the random potential. It
is this aspect we put special emphasis on in this contribution. \\
[3mm]
[1] \quad W.~Fischer, H.~Leschke, and P.~M\"uller: Towards
localization by Gaussian random potentials in multi-dimensional
continuous space, {\em Lett. Math. Phys.} {\bf 38}, 343--348 (1996).
\\ [0mm]
[2] \quad W.~Fischer, T.~Hupfer, H.~Leschke, and P.~M\"uller:
Rigorous results on Schr\"odinger operators with certain Gaussian
random potentials in multi-dimensional continuous space, in:
M.~Demuth and B.-W.~Schulze (eds.), {\em Differential equations,
asymptotic analysis, and mathematical physics,} Akademie Verlag,
Berlin 1997; pp.~105--112.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf The shifted--$l$ expansion technique to get eigenvalues of
Schr\"odinger, Dirac, and Klein--Gordon wave equation} \\ [5mm]
{\large O.~Mustafa} \\ [2mm]
University of Northern Cyprus, Famagusta
\end{center}
\noindent
The shifted--$l$ expansion technique (SLET) has been developed to get
eigenvalues of Schr\"odinger equation in three (3D) and two
dimensions (2D). SLET simply consists of $1/\bar{l}$ as a perturbation
parameter. Where $\bar{l} = l - \beta$, $\;\beta$ is a suitable shift, $l$
is the angular momentum quantum number for the 3D--case, $l = |m|$ for the
2D--case, and $m$ is the magnetic quantum number. SLET has been extended
to solve for Dirac equation with scalar and/or 4--vector potentials. A
parameter $\lambda = 0,1$ has been introduced in Dirac equation in such
a way that one can obtain Klein--Gordon bound states from those of Dirac.
SLET has been applied to some spherically symmetric potentials of interest.
Highly accurate and fast converging eigenvalues have been obtained.\\
[3mm]
1- ``Nonrelativistic Shifted--$l$ Expansion Technique for Three- and
Two-Di\-men\-sional Schr\"odinger Equation". Omar Mustafa and Thabit
Barakat, {\em Commun. Theor. Phys.} {\bf 28}, 257 (1997) \\
2- ``Relativistic Shifted--$l$ Expansion Technique for Dirac and
Klein--Gordon Equations" Omar Mustafa and Thabit Barakat, {\em
Commun. Theor. Phys.} (accepted for publication). \\
3- ``Perturbed Coulomb potentials in the Klein--Gordon equation via
the shifted--$l$ expansion technique". Thabit Barakat, Maen Odeh, and
Omar Mustafa, {\em J. Phys.} {\bf A}: Math. Gen. (accepted for
publication).
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Operator--norm convergence the Trotter--Kato product
formula} \\ [5mm]
{\large \underline{H.~Neidhardt}, V.A.~Zagrebnov} \\ [2mm]
FB Mathematik der TU Potsdam, PF 601553, D--14415 Potsdam
\end{center}
\noindent
Usually the Trotter-Kato product formula is considered in the strong
operator topology. In the last years, however, the problem arises to
verify the convergence in the operator-norm topology. We give
necessary and sufficient conditions for the convergence of the
Trotter-Kato product formula in the operator-norm topology. We prove
that compactness, relative compactness and product compactness
assumptions imply operator-norm convergence. Moreover, in application
to the Schr\"odinger operator it turns out that operator-norm
convergence is generic.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Nonexponential estimates of the angle between stable and
unstable separatrices for the Taylor--Chirikov--Green mapping} \\ [5mm]
{\large M.~Novitskii} \\ [2mm]
Institute for Low Temperature Physics, \\ Lenin Ave 47, 310164 Kharkov,
Ukraine
\end{center}
\noindent
We develop the Sinai-Cornfeld approach to the problem of the
estimate of the angle between separatrices at homoclinic point
for the Taylor-Chirikov- Green mapping. The nonexponential estimates
of this angle are given for the case when a function which generates
this mapping is not real analytic. Relations with the homoclinic
chaos and the homoclinic invariants are discussed.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Bell's inequality and \\ the Pauli exclusion principle} \\
[5mm]
{\large P.~O'Hara} \\ [2mm]
Department of Mathematics, North Illinois University, \\
5500 North St. Louis Ave, Chicago IL 60625--4699, U.S.A.
\end{center}
\noindent
Bell's inequality has ben traditionally used to explore the
relationship between hidden variables and the Copenhagen
interpretation of Quantum Mechanics. In this presentation another
use is found. By adapting an argument of Wigner, Bell's inequality is
first used to derive a coupling principle for elementary particles
with spin. Following this, the Pauli Exclusion Principle is derived
as a special case of the coupling principle. The meaning of the
spin-statistics theorem is then discussed.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Quantum conductance of molecular wires: Green function
description of real systems} \\ [5mm]
{\large A.~Onipko} \\ [2mm]
Bogolyubov Institute for Theoretical Physics, 252143 Kiev, Ukraine
\end{center}
\noindent
In our recent development of the Green function approach, Refs. 1-7
among other results we have formulated a simple physical model of
conjugated oligomers that (i) reasonably describes the $\pi$
electronic structure of this wide class of organic molecules, i.e.,
gives a quick and reliable estimate of band gaps and band widths,
transition dipoles, etc.; and (ii) provides an efficient tool of
in-depth theoretical understanding of optical and charge-transfer
properties of these promising compounds. Our analytical description
is complementary to (dominating) computational approaches and, in a
sense, represents an alternative and challenge to these.
%
The above mentioned model has been applied to the description of
quantum conductance of metal/molecular heterojunctions. It is
strictly proved for the first time that through band-gap tunneling,
that is believed to be the principle mechanism of the electrical
current in metal/molecular heterojunctions, is ruled by an
exponential dependence. (So far, this kind of dependence was either
guessed or found numerically.) A model-exact explicit definition of
the exponential decay constant and pre-exponential factor (in the
standard expression of tunnel conductance) is given in terms of the
Green function of the monomer -- structural unit of oligomers. This
makes possible to address the essentials of electron transfer across
large molecules at the monomer level and greatly facilitates
estimating the electron-transferring abilities of various kinds of
conjugated oligomers.
%
Using different techniques several groups have succeeded to observe
the current through a single molecule. The controllable charge
transport in molecular nanodevices is becoming the reality. From this
perspective we are intended to highlight the following issues: (i)
the relationship between the electronic structure of molecular wires
and the metal/molecular heterojunction resistance: (ii) inherent
switching abilities of conjugated oligomers including those that are
due to in-gap localized states; and (iii) resonances (due to in-band
states) and pseudo-resonances (due to in-gap states) in the
through-molecule transmission of electrons. Neither of these `hot'
topics has received yet an adequate coverage in the literature. \\
[1.5mm]
1. B. Kohler et al., J. Chem. Phys. {\bf 103}, 6068
(1995). \\
2. L. Malysheva and A. Onipko, Synthetic Metals {\bf 80}, 11 (1996).
\\
3. L. Malysheva and A. Onipko, J. Chem. Phys. {\bf 105}, 11032
(1996). \\
4. A. Onipko et al., J. Chem. Phys. {\bf 107}, 5032 (1997). \\
5. A. Onipko et al., J. Chem. Phys. {\bf 107}, 7331 (1997). \\
6. A. Onipko et al., J. Luminescence {\bf 76-77}, 658 (1998). \\
7. A. Onipko and Yu. Klymenko, J. Phys. Chem. A {\bf 102}, issue 81
(1998).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf The three-body Coulomb problem in three-potential
formalism} \\ [5mm]
{\large Z.~Papp} \\ [2mm]
Institute of Nuclear Research of the Hungarian Academy of Sciences,
\\ P.O. Box 51, H-4001 Debrecen, Hungary
\end{center}
\noindent
Herein, I propose a new approach to the three--body Coulomb
scattering. As usual, one presumes that the quantum mechanical
system evolves from a state governed by the asymptotic Hamiltonian to
the physical state described by the total Hamiltonian. In the
two--potential formalism of that, an intermediate Hamiltonian must
be defined and connection is made first to the asymptotic Hamiltonian
and then to the total Hamiltonian. In a ``three-potential'' picture
a three-body Coulomb scattering process can be viewed as three
consecutive scattering processes by which the asymptotic channel
Hamiltonian is connected to the total one via two intermediate
Hamiltonians. In the first step the system evolves from the state
described by the asymptotic channel Hamiltonian to the state
described by the channel-distorted Coulomb Hamiltonian. This is a
two-body Coulomb scattering. Then the system goes over to the state
governed by the channel Coulomb Hamiltonian, which contains all the
long-range potential terms. On this level there are no rearrangement
channels, so a single Lippmann--Schwinger equation, which possesses
compact kernel, guarantees for the unique solution. Finally, the link
is made to the total Hamiltonian. This is framed in terms of Faddeev
integral equation which contains only short-range type source terms.
So, the three-body Coulomb problem is described by a set of
Lippmann--Schwinger and Faddeev integral equations which possess
compact kernels for all energies.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Temporal evolutions in quantum mechanics and quantum Zeno
effect} \\ [5mm]
{\large \underline{S.~Pascazio}, P.~Facchi} \\ [2mm]
Dipartimento di Fisica and INFN, Universit\`a di Bari, Bari, Italy
\end{center}
\noindent
The temporal behavior of an unstable quantum system is analyzed in a
field-theoretical framework. We concentrate our attention on the
2P-1S transition of the hydrogen atom and scrutinize the deviations
from the exponential law: the ``quantum Zeno" time is found to be
approximately $3.59 \cdot 10^{-15}$s (the lifetime is approximately
$1.595 \cdot 10^{-9}$s) and the transition to a power law occurs
after 98 lifetimes.
We then consider Van Hove's weak-coupling ``$\lambda^2 t$" limit. In
this case, one obtains an exponential law at all times and a rigorous
approach to the Weisskopf-Wigner approximation. The direct comparison
with the exact formulae is very instructive and yields insight into
the issue of dissipation and irreversibility in quantum mechanics.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Some examples of inverse Sturm-Liouville problem with
three spectra} \\ [5mm]
{\large V.~Pivovarchik} \\ [2mm]
Department of Higher Mathematics, Odessa State Academy of Civil
Engineering and Architecture, Didrihson Str.~4, 270045 Odessa, Ukraine
\end{center}
\noindent
Inverse problems of the following form are considered: three spectra
are given one of which corresponds to a boundary problem (Dirichlet or
Dirichlet-Neumann problem) generated by the Sturm-Liouville equation
on a finite interval $[0,a]$ and the two others correspond to the
Dirichlet boundary problems on the subintervals $[0,b]$ and $[b,a]$.
The inverse problem lies in the construction of the potential from
the three spectra. The uniqueness of the solutions of such problems
is discussed. Some applications are given associated with the
construction of the density of a string from the spectra of its
vibrations. The case of partially damped string is also considered.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Solvable models for serially connected Aharonov--Bohm
rings and localization effects} \\ [5mm]
{\large I.~Popov} \\ [2mm]
Department of Higher Mathematics \\
Leningrad Institute of Fine Mechanics and Optics \\
14 Sablinskaya, 197101 Sankt Peterburg, Russia
\end{center}
\noindent
Two simple explicitly solvable models based on the operator extension
theory techniques are suggested. The class of the mentioned models
includes, in particular, the famous Kronig--Penney model of solid
state physics. The class is usefully employed for studying transport
properties of nanostructures [1]. We deal with two infinite chains
of the Aharonov--Bohm rings. One obtains the dispersion equations
for the model operators in an explicit form using Krein's resolvent
formula. It allows one to study the spectrum in details. In the first
geometrical configuration the rings are connected directly
one--to--one. It is shown that the spectrum consists only of
localized states under the condition that the number of magnetic flux
quanta through the ring is half--integer. In the second chain the
rings are laterally coupled with infinite wire. In this, more
realistic case, the localization condition includes a relation
between the circle radius and the length of the wire connecting two
neighbour rings. Besides, there are delocalized states in the
spectrum. The work was partly supported by RFBR. \\ [2mm]
1. V.A.Geyler, B.S.Pavlov, I.Yu.Popov, {\em J. Math. Phys.} 37
(1996), 5171.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Asymptotic properties of the ``magnetic" density of
states} \\ [5mm]
{\large G.D.~Raikov} \\ [2mm]
Institute of Mathematics and Informatics, Bulgarian Academy \\ of
Sciences, Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
\end{center}
\newcommand{\dl}{{\cal D}(\lambda )}
\newcommand{\lo}{\lambda \uparrow 0}
\newcommand{\llo}{\lambda \downarrow \lambda_0}
\newcommand{\leo}{\lambda \downarrow {\cal E}_0}
\noindent
Let $H(\mu ):= \left(i\nabla+ \mu A\right)^2 + V$ be the Schr\"odinger
operator. Here $A: {\bf R}^m \rightarrow {\bf R}^m$, $m \geq 3$, is the
magnetic potential associated with a constant magnetic field $B$ such
that $k:= {\rm dim}\, {\rm Ker}\,B \geq 1$, $V: {\bf R}^m \rightarrow
{\bf R}$ is the electric potential such that $V \leq 0$, $V \not
\equiv 0$, and for each $\varepsilon > 0$ we have $V= V_1 + V_2$ with
$V_1 \in L^{m/2}({\bf R}^m)$, $\sup_{X \in {\bf R}^m}|V_2(X)| \leq
\varepsilon$, and $\mu \geq 0$ is the magnetic-field coupling
constant. Hence, $\sigma_{\rm ess}(H(\mu)) = [\mu \Lambda, \infty)$,
$\Lambda: = {\rm Tr}\,\sqrt{B^*B}$ being the first Landau level.
Assume that the negative spectrum of a given selfadjoint operator $T$
is purely discrete, and denote by $N(\lambda;T)$ the number of its
eigenvalues smaller than $\lambda < 0$.
In a recent work [1] the author has derived the asymptotic formula
$$
\lim_{\mu \rightarrow \infty } \mu^{-d} N(\lambda ;H(\mu ) - \mu \Lambda) =
(2\pi)^{-d} \det B_+ \, {\cal D}(\lambda ), \; \lambda < 0,
$$
where $2d = {\rm dim}\,{\rm Ker}\, B = m-k$, $B_+$ is the restriction
of $\sqrt{B^* B}$ onto ${\rm Ran} \, B$, and
$$
\dl := \int_{{\bf R}^{2d}} N(\lambda; \chi(X_\perp))\,dX_\perp, \;
\lambda < 0,
$$
is the integrated magnetic density of states,
the operator $\chi(X_\perp):= -\Delta +V(X_\perp,.)$ acting in
$L^2({\bf R}^k)$, ${\bf R}^k = {\rm Ker}\,B$,
and depending on the parameter $X_\perp \in {\bf R}^{2d} = {\rm Ran}\,B$.
Assume that ${\cal D}$ does not vanish identically. Set $\lambda_0 :=
\inf {\rm supp}\; {\cal D}$. The asymptotic properties of $\dl $ as
$\lo $, or as $\llo$, are discussed in the talk.
First, under the assumption that $V$ admits a power-like decay at
infinity, we analyze the asymptotic behaviour of $\dl$ as $\lo$.
Further, we assume $\lambda_0 > -\infty$, and reduce the
investigation of the asymptotics as $\llo$ of $\dl$ to a relatively
standard problem from the continuous perturbation theory for
selfadjoint operators.
Finally, in the case $m=3$, we introduce a class of asymptotically
homogeneous potentials including the Coulomb one, for which
$\lambda_0 = -\infty$, and study the asymptotics of $\dl $ as
$\lambda \rightarrow -\infty$. \\ [3mm]
1. {\sc G.D.Raikov}: Eigenvalue asymptotics for the Schr\"odinger
operator in strong constant magnetic fields, to appear in {\em
Commun. P.D.E.}
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Symmetries of a completely integrable Hamiltonian system}
\\ [5mm]
{\large L.~Remezo} \\ [2mm]
Service de Physique Theorique et Mathematique, Universite de
Li\`{e}ge \\ B--4000 Li\`ege 1, Belgium,
and University of Burundi, Bujumbura
\end{center}
\noindent
We put in evidence the symmetries associated with the {\it quantum}
versions of a completely integrable Hamiltonian recently introduced by
Calogero. The corresponding algebra is seen to be $sp(2;\R)$.
\vspace{25mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Embedded singular spectrum of \\ 1D Schr\"odinger operators}
\\ [5mm]
{\large Ch.~Remling} \\ [2mm]
FB Mathematik der Universit\"at Osnabr\"uck, D--49069 Osnabr\"uck
\end{center}
\noindent
Consider the one-dimensional Schroedinger
equation $-y''(x) + V(x)y(x)=Ey(x)$. We will discus
results which imply presence of absolutely continuous
spectrum on $(0,\infty)$ if $V$ is asymptotically
small (a typical assumption is $V(x)=O(x^{-c}$ with
$c>1/2$). In general, embedded singular spectrum can
occur. We will present further results and examples
related to these issues.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Limiting absorption principle for \\ singularly perturbed
operators} \\ [5mm]
{\large W.~Renger} \\ [2mm]
Institut f\"ur Mathematik, TU Clausthal, D--38678
Clausthal--Zellerfeld \\ {\em mawr@tu--clauthal.de}
\end{center}
\noindent
Suppose a limiting absorption principle (LAP) holds for a
self-adjoint, semibounded operator $H_1$ in $\HH_1$. That is, suppose
there is a dense subspace $X\subset\HH_1$ and an open set
$\Delta\subset\R$ such that $R_1^{\pm}(\lambda):=
\lim_{\varepsilon\downarrow 0} R(\lambda\pm i\varepsilon)$ exists in
the norm topology of $\BB(X^*,X)$ (the space of bounded operators
from $X$ to $X^*$) for all $\lambda\in\Delta$.
We study operators $H_2$ which are produced be perturbing $H_1$ in
some sense. Primarily we are interested in singular perturbations
where the operator difference $H_2-H_1$ is meaningless. We suppose
that the perturbation is small in the sense that the difference
between some powers of the resolvents $R_j(a)$ is compact: $R_2(a)^m
-R_1(a)^m \in\BB_{\infty}(X^*,X)$ for some $m\in\N$. On this abstract
level we show that (except for possibly a discrete set of
eigenvalues) a LAP holds for $H_2$.
We apply this theory to study potential and domain perturbations of
Feller operators (operators which are defined as generators of strong
Markov processes with the Feller property). We assume that a LAP
holds for the unperturbed operator $H_1$ on $L^2(M)$ ($M$ some
appropriate measure space) and consider the perturbed operator $H_2=
(H_1+V)_\Sigma$ that is produced by restricting $H_1+V$ to a set
$\Sigma \subset M$ via Dirichlet boundary conditions. We give
sufficient conditions in terms of $V$ and the equilibrium potential
of the set $\Gamma=M\setminus\Sigma$ to ensure that a limiting
absorption principle holds for $H_2$. If, for instance, $H_1$ is the
Laplacian our theory allows to treat the usual short range
potentials, but it also allows domain perturbations by sets $\Gamma$
which may be unbounded -- the condition we impose on $\Gamma$ is
slightly stronger than requiring that its capacity is finite.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Spectral shift function for the Dirac operator} \\ [5mm]
{\large D.~Robert} \\ [2mm]
Departement de Math\'ematiques, Universit\'{e} de Nantes, \\
2, rue de la Houssiniere, F--44322 Nantes
\end{center}
\noindent
Perturbations of the free Dirac operator on $\R^n$ by electromagnetic
fields are considered. For these perturbations we can prove
asymptotic expansions for the spectral shift function in different
regimes: small Planck constant, high energy, nonrelativistic limit.
We also discuss low energy behaviour, Levinson type formulas and
trace identities.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Eigenvalue estimates for Schr\"odinger--like operators
with magnetic field} \\ [5mm]
{\large G.~Rozenblioum} \\ [2mm]
Department of Mathematics, G\"oteborg University, \\ SE--41296
G\"oteborg, Sweden
\end{center}
\noindent
It is well known that the semigroup generated by the Schr\"odinger
operator with magnetic field is dominated by the similar
Schr\"odinger semigroup without magnetic field. This, however, does
not imply corresponding domination for eigenvalues. We present an
approach to obtaining CLR--type eigenvalue estimates which enables
one to derive them from semigroup domination. This, in particular,
gives versions of the CLR--bound for certain other operators
involving magnetic field, in particular the relativistic
Schr\"odinger operator and the Dirac operator.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf One--dimensional models for many--electron \\ atoms in
strong magnetic fields} \\ [5mm]
{\large R.~Brummelhuis, \underline{M.B.~Ruskai}} \\ [2mm]
Department of Mathematics, University of Massachussets at Lowell, \\
Lowel MA 01854, U.S.A.
\end{center}
\noindent
We consider a one-dimensional model for many-electron atoms in
strong magnetic fields in which the Coulomb potential and
interactions are replaced by one-dimensional regularizations
associated with the lowest Landau level. For this model we show that
the maximum number of electrons $N$ satisfies the bound $N <
2Z+1+B/2$ where $Z$ denotes the charge of the nucleus and $B$ denotes
the field strength. We follows Lieb's strategy in which convexity
of the potential plays a critical role.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Generic behavior of \\ topological quantum numbers} \\
[5mm]
{\large L.~Sadun} \\ [2mm]
Mathematics Department C1200, University of Texas, \\ Austin TX 78712,
U.S.A.
\end{center}
\noindent
Topological quantum numbers are frequently given by the indices of
Fredholm operators. As the parameters of the system are varied,
these indices can change (discontinuously, of course). We say a
family of operators exhibits generic behavior if the operators are
Fredholm almost everywhere, with the indices jumping by 1 on a set of
codimension 1, by 2 on a set of codimension 2, and so on. Any finite
set of Fredholm operators may be interpolated by a generic family,
and generic behavior occurs in a wide class of physically relevant
systems. However, there are important classes of operators that are
not generic, and we present criteria for distinguishing generic from
non-generic. (This is joint work with J.E. Avron).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf The discrete spectrum in the gaps of the continuous one
for sign indefinite perturbations with a large coupling constant} \\
[5mm]
{\large O.~Safronov} \\ [2mm]
Department of Mathematics, Royal Institute of Technology\\
S--10044 Stockholm, Sweden; {\em e-mail:} safronov@@math.kth.se
\end{center}
\noindent
Given two selfadjoint operators $A$ and $V=V_+-V_-$, we study the
motion of the eigenvalues of the operator $A(t)=A-tV$ as $t$
increases. Let $\alpha>0$ and let $\lambda$ be a regular point for
$A$. We consider the quantities $N_+(\lambda,\alpha),\
N_-(\lambda,\alpha),\ N_0(\lambda,\alpha)$ defined as the number of
the eigenvalues of the operator $A(t)$ that pass point $\lambda$ from
the right to the left, from the left to the right or change the
direction of their motion exactly at point $\lambda$, respectively,
as $t$ increases from $0$ to $\alpha>0.$ An abstract theorem on the
asymptotics for these quantities is presented. Applications to
Schr\"odinger operators and its generalizations are given.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf KAM theory for quantum systems} \\ [4mm]
{\large W.~Scherer} \\ [2mm]
Institut f\"ur Theoretische Physik A, TU Clausthal \\
Leibnizstr. 10, 38678 Clausthal--Zellerfeld, Germany
\end{center}
\noindent
Formulating the KAM algorithm for classical Hamiltonian systems in a
geometric fashion and using the fact that the Schr\"odinger equation
can be written as infinite--dimensional Hamiltonian system, general
formal analogues of the KAM algorithm both for time independent as
well as for time dependent systems are constructed.
\vspace{4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Quantum--mechanical symmetries and self--adjoint
extensions on the pointed plane} \\ [4mm]
{\large Ch.~Schulte} \\ [2mm]
Arnold Sommerfeld Institut, TU Clausthal, D--38678
Clausthal--Zellerfeld
\end{center}
\noindent
On the two-dimensional pointed plane, the quantization of a classical
system is not unique and -- in addition -- the resulting Hamiltonian
defined on the set smooth functions with compact support is only
symmetric, but not essentially self-adjoint.
%
In a first step, it will be shown briefly how its self-adjoint
extensions for a quantum mechanical system evolving in a $\frac
1r$-potential look like. Although all self-adjoint extensions are
equally justified from a mathematical point of view, they exhibit
particularities for the quantized symmetries of the classical system:
this will be investigated in the second step and it will turn out
that only for special quantization mappings and special self-adjoint
extensions the quantized symmetries lead to degeneracies of the
eigenspaces. In all other situations the eigenspaces of the
Hamiltonian are not invariant under the quantized symmetries and it
is not possible to define the symmetry operators and the Hamiltonian
on a common dense, invariant domain that contains the eigenfunctions
of the Hamiltonian.
\vspace{4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Resonance trapping in a weakly open quantum dot} \\ [4mm]
{\large P.~\v{S}eba} \\ [2mm]
Nuclear Physics Institute, Academy of Sciences, \v{R}e\v{z}, and \\
Doppler Institute, Czech Technical University, Prague
\end{center}
\noindent
We show that increasing the coupling between the bound
states and continuum can under certain conditions stabilize
the system and create states with extremely long lifetimes.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Vortices in Ginzburg--Landau equation: \\ numerical
results and analytical problems}
\\ [5mm]
{\large V.V.~Kurin, I.M.~Nefedov, \underline{I.A.~Shereshevsky},
P.P.~Vysheslavtsev} \\ [2mm]
Institute for Physics of Microstructures, RAS,
Nizhni Novgorod, Russia \\
{\em e-mail: ilya@ipm.sci-nnov.ru}
\end{center}
\noindent
We consider a mathematical model of the dynamics of magnetic
vortices in a small superconducting film in external electric and
magnetic fields. In such case the dynamics of a film is described by
the time-dependent two dimensional Ginsburg-Landau equation with the
electric potential depended on super- and external current. These
equations are added with boundary conditions for the electric
potential and order parameter.
One of the main features of this model is the appearance of the {\it
vortices}, i.e., isolated zeros of order parameter, optionally moving
in time. Due to many physical and mathematical reasons the
investigation of the vortex structures and dynamics is the main
goal of the qualitative theory of described model.
As the first step for numerical investigation of the vortices one
must construct the difference variant of the Ginzburg-Landau
differential model. Such a model must satisfy some natural
conditions to exclude the artificial phenomena of `discreteness'. One
of them is gauge invariance of the model and another one is the
absence of the electric field in the stationary state at zero
external current. We describe such difference model and prove its
above mentioned properties. We also introduce the difference variant
of the notion of `vortex' and the method for recognizing of the
vortex positions and traces. We demonstrate some interesting features
of vortex configuration and dynamics obtained in numerical
experiments.
Although there exists a very large bibliography concerning the
numerical investigation of the different Ginzburg-Landau models,
there are only scarce mathematical results on this problem.
One of the essential problem of the interpretation of numerical
results connects with the effect of `grid pinning' of the vortices
in discrete model. So it's important to give the analytical
determination of a `pinning force' for distinguishing the grid
depending effects from the natural ones.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Conductance of interacting region \\ attached to
noninteracting leads} \\ [5mm]
{\large K.~Shundyak} \\ [2mm]
Department of Theoretical Physics, University of Odessa, \\
Dvoryanskaya 2, 270000 Odessa, Ukraine
\end{center}
\noindent
The conductivity of a quantum region locking two quantum wires is
calculated on the base of the Landauer formula with account of direct
interaction and identity of carriers. Variations of the obtained
relation under impact of external electric and magnetic fields are
studied. The main results are generalized for cases where a quantum
region locks more than two quantum
wires.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Mathematics of quantum computing} \\ [5mm]
{\large G.~Sobczyk} \\ [2mm]
Departamento de Fisica y Matematicas, Universidad de los Americas, \\
Cholula, Puebla 72820, Mexico
\end{center}
\noindent
The idea that it is possible to construct a new type of computer
based on Quantum Mechanics has opened the door to a New Science and
a reexamination of natural processes. We closely study the mathematical
basis of many particle spin states using Clifford algebra and extended
factor algebras of polynomials. \\[3mm]
1. D. Hestenes, Observables, operators and complex numbers in the Dirac
theory, {\em JMP}, Vol. 16, No. 3, 1975, pp. 556-572 \\
2. D.P. DiVincenzo, Quantum Computation, {\em Science}, Vol. 270,
1995, pp.255-260. \\
3. S.S. Somaroo, D.G. Cory, T.F. Havel, Expressing the operations of
quantum computing in multiparticle geometric algebra, 1998 (to appear) \\
4. G. Sobczyk, Spectral integral domains in the classroom, {\em
Aportaciones Matematicas}, Vol. 20, 1997, pp.169-188. \\
5. ----------, Quantum Computing, Nuclear Magnetic Resonance and Clifford
Algebra, (Submitted to {\em Aportaciones Matematicas} 1998).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf A reverse log--Sobolev inequality \\ in the
Segal--Bargmann space} \\ [5mm]
{\large S.B.~Sontz} \\ [2mm]
Departamento de matematicas, Universidad Autonoma
Metropolitana-Iztapalapa \\ Col. Vicentina, Mexico DF 09340,
Mexico
\end{center}
\noindent
The Segal-Bargmann space (of entire analytic functions
on ${\bf C}^n$ which are square integrable with respect to a
Gaussian measure) arises as a Hilbert space adequate for doing
standard quantum mechanics, since it supports an irreducible
representation of the canonical commutation relations, i.e.
the Weyl group. As a Hilbert space of entire functions, the
Segal-Bargmann space also has a reproducing kernel function
which, in turn, has an associated integral kernel transform.
The mapping properties of this transform are identified with
respect to the $L^p$ scale of spaces. Then by using a differention
technique of Hirschman, this leads to a family of log-Sobolev
inequalities in the Segal-Bargmann space. While it is already
well known that there is such an inequality which gives an
upper bound of the entropy in terms of the energy (of the
harmonic oscillator), we show that this family of inequalities
also includes a reverse type inequality, namely an upper
bound of the energy in terms of the entropy. It follows that
the entropy is finite if {\it and only if} the energy is finite.
These results are all proved without using hypercontractivity
estimates.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Hydrodynamics for quasi--free \\ quantum systems} \\ [5mm]
{\large W.~Spitzer} \\ [2mm]
Department of Physics, Princeton University, Princeton NJ 08544
\end{center}
\noindent
We consider quasi-free quantum systems and derive the
Euler-equation using the so-called hydrodynamic limit. We use Wigner's
well-known distribution function and discuss an extension to band
distribution functions for particles in a periodic potential. Further, we
investigate the Bosonic system of hard rods and calculate fluctuations of
the density.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Multiparameter spectral averaging and localization for the
random displacement model} \\ [5mm]
{\large G.~Stolz} \\ [2mm]
Department of Mathematics, University of Alabama, \\ Birmingham AL
35294--2154, U.S.A.
\end{center}
\noindent
The method of spectral averaging, going back to works of Carmona, Kotani,
and Simon in the 1980s, is one of the main tools in proofs of localization
for random Schrodinger operators. So far, the applicability of spectral
averaging was limited to models with monotonous dependence on the random
parameters.
By introducing the idea of multiparameter spectral averaging, i.e. by
varying several independent random parameters simultaneously, we are able
to treat a number of physically important models with non-monotonous
parameter dependence. A key observation is that monotonicity can be
replaced by analyticity as long as non-constancy is guaranteed, where the
latter invokes some input from inverse spectral theory.
Applications of these results include proofs of localization at all
energies and without sign restrictions on the single site potential for all
the following one-dimensional continuous random Schrodinger operators: (i)
The random displacement model (where no previous results were known), (ii)
the Poisson model (previously only treated at positive energy), and (iii)
Anderson-type models (previously treated for single site potentials with
fixed sign).
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Perturbation of an eigen-value from a dense point
spectrum: a general Floquet Hamiltonian} \\ [5mm]
{\large P.~Duclos, \underline{P.~\v{S}\v{t}ov\'{\i}\v{c}ek},
M.~Vittot}
\\ [2mm]
Department of Mathematics, FNSPE, Czech Technical University, \\
Trojanova 13, 12000 Prague
\end{center}
\noindent
We consider a perturbed Floquet Hamiltonian
$-i\partial_t + H+\beta V(\omega t)$ in the Hilbert space
$L^2([0,T],\HH,dt)$. Here $H$ is a self-adjoint operator in
$\HH$ with a discrete spectrum obeying a growing gap condition,
$V(t)$ is a symmetric bounded operator in $\cal H$
depending on $t$ $2\pi$-periodically,
$\omega=2\pi/T$ is a frequency and
$\beta$ is a coupling constant. The spectrum $\sigma(-i\partial_t+H)$
of the unperturbed part is pure point and dense in
{\bf R} for almost every $\omega$. This fact excludes application
of the regular perturbation
theory. Nevertheless we show, for almost all $\omega$ and provided
$V(t)$ is sufficiently smooth, that the perturbation theory still
makes sense, however, with two modifications. First, the coupling
constant is restricted to a set $I$ which need not be an interval but
0 is still a point of density of $I$. Second, the
Rayleigh-Schr\"odinger series are asymptotic to the perturbed
eigen-value and the perturbed eigen-vector.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Solving the Schr\"odinger equation with non-singular
potentials using Hamiltonian degrees} \\ [5mm]
{\large \underline{A.I.~Streltsov}, I.A.~Morev, I.P.~Bartashevich,
D.V.~Podgainy} \\ [2mm]
Physics Department, Far East State University, \\
Sukhanova St. 8, 690600 Vladivostok, Russia
\end{center}
\noindent
The Method of Connection Moments [1,2] has been applied to
calculations of the ground-state of the Schrodinger equation.
The excited states were calculated by means of the Hamiltonian degrees.
This approach is also useful for the calculation of the ground-state.
The convergence of the formalism was proved using Pade-Approximations.
The method may be presented in a non-integral form.
In this case we have investigated dependence of the eigenvalues on
the representing point and parameters of an arbitrary initial function.
The formalisms was applied to model potential of the following kind:
$V(x)=kx^2+mx^4+gx^6$. The energy eigenvalues, given by the methods,
were in good agreement with the results of the standard Rayleigh-Schrodinger
perturbation theory [3,4], and a weak dependence on the choice of
the initial function was found. For the non-integral method we have
also observed a weak dependence on the choice of the representation point.
\vspace{3mm}
\noindent
1. Cioslowski J., et al., {\em Chem. Phys. Lett.} {\bf 38} (1987),
516 \\
2. Cioslowski J., {\em Int. J. Quant. Chem.} {\bf 21} (1987), 563 \\
3. Kleinert H., Janke W., {\em Phys. Lett.} {\bf A 206} (1995), 283
\\
4. Janke W., Kleinert H., {\em Phys. Rev. Lett.} {\bf 75} (1995),
2787
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Two--dimensional periodic Pauli operator. Effective masses
at the lower edge of the spectrum} \\ [5mm]
{\large M.S.~Birman, \underline{T.A.~Suslina}} \\ [2mm]
Institute of Physics, Sankt Peterburg State University, \\
Ulyanovskaya 1--1, 198804 Sankt Peterburg, Russia
\end{center}
\noindent
Let a magnetic potential ${\bf A} \in C^1(R^2)$ be a periodic
vector-valued function such that ${\rm div}{\bf A}=0$, $\int_\Omega
{\bf A}(x)\,dx=0$, where $\Omega$ is an elementary cell of period
lattice. Let $B(x)= \partial_2 A_1 - \partial_1 A_2$ be a magnetic
field. The Pauli operator is studied
$$P=\left(\matrix{M_-& 0 \cr 0 & M_+}\right);\ \ M_\pm = \left(
-i\nabla - {\bf A}(x)\right)^2 \pm B(x). $$
An equation $M_\pm \omega=0$ has unique (up to multiplier) periodic
solution $\omega_\pm (x)$. Moreover, $00.
$$
\par
Consider the following real function:
$$
b(t)=3Dt^{{1 \over 2}} (\log (t+2))^{-r},
$$
where $r>1, \ d=3D1; \ \ r>3/4, \ d=3D2$, and
$$
b(t)=3Dt^{{2 \over d+2}} (\log (t+2))^{-r}, \ \ r>{2 \over d+2}, \ d\ge =
3.
$$
\par
We show that with probability 1
for any $\psi_0 \in l^2(\Z^d)$
for $T$ sufficiently large
$$
\langle \vert X\vert ^p \rangle _{T} \ge {1 \over 2}
\Vert \psi_0 \Vert ^2 b^p (T).
$$
We show also that for
any $\psi_0$ exponentially decreasing at infinity, with probability 1
$$
\langle \vert X \vert ^p \rangle _{T} \le C
T^{{p \over 2}} (\log T)^r,
$$
where $r>1$ for $d=3D1,2$ and $r>p+1$ for $d \ge 3$.
\par
We obtain also upper bounds for the autocorrelation function
$$
C_{\psi_0} (T)=3DT^{-1} \int_0^T dt
\vert (\psi (t), \psi_0 ) \vert ^2.
$$
We show that for any $\psi _0$ with probability 1
$$
\limsup_{T \to +\infty} {\log C_{\psi _0}
(T) \over \log T} \le -{1 \over 2},
\ \ \ d=3D1;
$$
$$
\lim_{T \to +\infty}
{\log C_{\psi _0}(T)\over \log T} =3D-1, \ \ d \ge 2.
$$
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Schr\"odinger equation with \\ concentrated nonlinearities}
\\ [5mm]
{\large G.F.~Dell'Antonio, R.~Figari, \underline{A.~Teta}} \\ [2mm]
Universita di Roma ``La Sapienza", Piazalle A.~Moro 5, I--00185 Roma
\end{center}
\noindent
We consider the Schr\"odinger equation for a particle in dimension
three subject to a nonlinear point interaction. Such model is simple
but nontrivial example of concentrated nonlinearity. In particular we
suppose that the strength of the interaction is nonlinearly coupled
to the wave function of the particle. We find sufficient conditions
for the global existence and the blow-up of the solution. The result
is compared with the standard theory of NLSE.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Quantisation of conservative \\ and lossy dielectrics} \\
[5mm]
{\large A.~Tip} \\ [2mm]
FOM Institute for Atomic and Molecular Physics, \\
Kruijslaan 407, NL--1098 Amsterdam, The Netherlands
\end{center}
\noindent
Starting from a unitary time evolution in a real separable
Hilbert space, a Lagrange-Hamilton formalism is developed. In case
the generator has a non-empty null space, a generalization of the
gauge concept makes its appearance, leading to abstract versions of
the familiar gauges of electrodynamics. After fixing the gauge the
Dirac recipe can now be used for the quantization of the system. The
application to conservative dielectrics is then immediate. Lossy
dielectrics, where the relation between the displacement D and
electric field $E$ is convolutive, $D(t)=E(t)+ (X*E)(t)$, where
$X(t)$ is the electric susceptibility and the * indicates a
convolution in time, can not directly be quantized. However, it is
possible, by introducing two additional real vector fields, to obtain
a coupled set of fields that once more shows a unitary time evolution
and which is equivalent to the original Maxwell's equations. This
formulation is useful for the definition of band gaps in a lossy
situation. Next the abstract procedure, mentioned above, can be used
for the quantization of the system. Among other matters, there is a
difference with the conservative situation in that now a quantum
Langevin force term makes its appearance. An important application is
the decay of atoms, embedded in dielectric materials.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Perturbation theory for boundary S--matrix in 2D quantum
field theory} \\ [5mm]
{\large N.~Topor} \\ [2mm]
Department of Physics, Union College, Schenectady NY 12305--3161
\end{center}
\noindent
We develop a perturbation theory for evaluating the boundary S-matrix
in 2D quantum field theory. We apply this approach to calculate the
one-loop boundary S-matrix for the elementary particle of the
sine-Gordon model with a boundary interaction. Our perturbative
result agrees with the exact expression of the S matrix conjectured
by Goshal and Zamolodchikov; it also allows us to derive the
perturbative relation between the free parameter theta in the
S-matrix and the free parameter in the boundary action, in the
particular case in which its other free parameter is zero.
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Bend--imitating approach to quantum wire--like
nanostructures} \\ [5mm]
{\large O.~Vakhnenko} \\ [2mm]
Bogolyubov Institute for Theoretical Physics,
Ky\"\i v, Ukra\"\i na \\ {\em e-mail: vakhnenko@gluk.apc.org}
\end{center}
\noindent
The bend-imitating models of singly- and multiply-bent $2D$ and $3D$
quantum wires with constant cross-section are proposed [1]. The
planar model is generalized to the needs of quantum wires subjected to
perpendicular magnetic field [2]. According to our approach each
circular-like bend of continuous quantum wire can be treated as some
multichannel point scatterer with scattering ability depending on the
bending angle, bending radius, magnetic field and time (or electron
energy in a steady-state regime). The problem of bend-caused
electronic localization is studied and the delocalizing effect of a
magnetic field is predicted [2]. The energies of localized electronic
states and their dependencies on geometrical parameters for some $2D$
and $3D$ wire-like structures without magnetic field are analytically
calculated [3,4,1]. Within the $2D$ bend-imitating model strict but
rather simple analytical conditions for the full electronic
transparency of quadruply-bent wire-like structures are formulated in
terms of electron energy and geometric parameters of the structure
[1]. \\ [2mm]
1. O.O. Vakhnenko, {\em Phys. Lett.} A {\bf 231}, 412 (1997). \\
2. O.O. Vakhnenko, {\em Phys. Lett.} A (submitted).\\
3. O.O. Vakhnenko, {\em Phys. Rev.} B {\bf 52}, 17386 (1995). \\
4. O.O. Vakhnenko, {\em Phys. Lett.} A {\bf 211}, 46 (1996). \\
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf The Segal--Bargmann transform \\ and canonical
transformations} \\ [5mm]
{\large C.~Villegas--Blas} \\ [2mm]
Universidad Nacional Autonoma de Mexico, Instituto de Matematicas, \\
Unidad Cuernavaca, Apartado Postal 273-3 Admon. 3 \\ Cuernavaca
Morelos 62251, MEXICO
\end{center}
\noindent
The usual Segal-Bargmann Transform (SBT) is a Unitary operator going from
the Hilbert space of square-integrable functions in n real variables onto
the Hilbert space (Bargmann space) of analytic and square-integrable
functions in n complex variables with respect to a Gaussian measure. It is
known that the SBT can be written as an integral operator whose kernel is a
power series of a generating function of a linear canonical transformation
(CT) which gives the classical version of what the SBT does in quantum
theory. In particular, the SBT intertwines the Hamiltonian of n harmonic
oscillators with Hamiltonians which are equal to a creation operator
(multiplication by the complex variable z) times an annihilation operator
(derivative with respect to z) whereas (via the CT) the harmonic
oscillators appear (in classical mechanics) as the addition of operators of
the form "z times z complex conjugate".
The main goal of the talk is to show that the mentioned parallelism
between the SBT and a corresponding canonical transformation holds in two
more cases. Namely, when the configuration space is the Hilbert space of
square-integral functions on the 2-sphere and the 3-sphere with respect to
the usual surface measure. In the case of the 2-sphere we use as a SBT
the one introduce by L.Thomas and S. Wassell and for the 3-sphere case we
introduce a SBT taking into account the symmetries of the hydrogen atom
problem. In both cases the CT is non-linear. The Hamiltonian we analyze
in both cases is the one of a particle moving on the sphere (in quantum
and classical mechanics). In the 2-sphere case, this Hamiltonian appears,
in the Bargmann space, as the square of two harmonic oscillators and in
the 3-sphere case as the addition of four harmonic oscillators plus a
restriction.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf The non--autonomous Kato class}
\\ [5mm]
{\large J.~Voigt} \\ [2mm]
Technische Universit\"at Dresden, Fachrichtung
Mathematik, \\ D--01062 Dresden, Germany
\end{center}
\noindent
We discuss singular time dependent absorption--excitation rates for
the heat equation. The {\it non--autonomous enlarged Kato class} is
defined by
$$\widehat{\bf NK}:=\{V:[0,\infty)\times\R^n\to \C\: {\rm
measurable}; \|V(\cdot)\|_U<\infty\},$$
where
$$\|V(\cdot)\|_U:=\sup_{s\ge0}\sup_{f\in D(\Delta),\,\|f\|_1\le1}
\int_0^1\|V(s+t)U(t)f\|_1\,dt, $$
and $U(\cdot)$ is the $C_0$-semigroup on $L_1(\R^n)$ generated by $\Delta$.
For $V \in \widehat{\bf NK},$ $\|V(\cdot)\|_U < 1,$ the initial value
problems
\begin{eqnarray*}
\frac{\partial}{\partial t} u(t,x) & = & \Delta u(t,x) + V(t,x)u(t,x),\\
u(s,x) & = & f(x), \quad t\ge s,\;x\in\R^n,
\end{eqnarray*}
have unique mild solutions in $L_1(\R^n)$. This is a consequence of the
non--autonomous
Miyadera perturbation theorem (cf. [1]). Sufficient conditions for
$V$ to belong to $\widehat{\bf NK}$ can be obtained in terms of
integrability
conditions and in terms of the usual Kato class (cf. [2]). \\[3mm]
1. F. R\"abiger, A. Rhandi, R. Schnaubelt,
J. Voigt: Non--autonomous \newline Miyadera perturbations, \rm submitted.
\\
2. R. Schnaubelt, J. Voigt: The non--autonomous Kato class, \rm
submitted.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Some applications of the ${\cal H}_{-2}$--construction} \\
[5mm]
{\large K.~Watanabe} \\ [2mm]
Department of Mathematics, Gakushuin University, Tokyo 171, Japan
\end{center}
\noindent
In this talk we consider the self-adjoint operators defined by the
$\HH_{-2}$--con\-stru\-ction (A.~Kiselev and B.~Simon, J. Funct. Anal
{\bf 130}, 345--356 (1995), and in the general case S.T.~Kuroda and
H.~Nagatani (see the corresponding plenary talk)). We give the
results related to the positive eigenvalue and the scattering theory
of the self-adjoint operators by using the operator-theoretical
approach, and apply the results to the Schr\"odinger operators in
$\R^3$ with the point-, line-, and surface-interactions.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Another look at Cwikel's inequality} \\
[5mm]
{\large T.~Weidl} \\ [2mm]
Matematiska Institutionen, Kungliga Tekniska Hoegskolan, \\
S-10044 Stockholm, Sverige
\end{center}
\noindent
Cwikel's bound [1]
%\cite{[C]}
on the singular numbers $s_k$ of the integral operators
$a(x)b(D)$ is a valuable tool to derive sharp uniform spectral
bounds, see, e.g., the celebrated Cwikel-Lieb-Rosenblum inequality.
In [2]
%\cite{[W]}
I pointed out, that a careful analysis of Cwikel's
proof implies, that the inclusion of $h(x,\xi)=a(x)b(\xi)$ to some
K-interpolation
space between $L_2({\R}^n\times{\R}^n)$ and
$L_\infty({\R}^n\times{\R}^n)$
implies, that $a(x)b(D)$ belongs to the
parallel interpolational operator ideal.
In applications to non-homogeneous symbols $a(x)b(\xi)$
with non-trivial dependence on additional parameters it turns out to
be useful to take
also the ``undressed'' version of these
estimates into account; namely
%
\[s_k\leq C(n)\left(\frac{1}{(2\pi)^d k}\int_0^{(2\pi)^n k}
(h^*(\tau))^2d\tau
\right)^{1/2}\]
%
where $h^*$ is the non-increasing rearrangement of $h$. \\[3mm]
%\begin{thebibliography}{99}
%\bibitem{[C]}
1. M. Cwikel: Weak type estimates for singular values and the number
of bound states of Schr\"odinger operators,
{\em Ann. of Math.}, {\bf 106} (1977) 93--100. \\
% \bibitem{[W]}
2. T. Weidl: Cwikel type estimates in nonpower ideals,
{\em Mathematische Nach\-rich\-ten}, {\bf 176} (1995) 315--334.
\vspace{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf The discrete spectrum in the singular Friedrichs model
with oscillating integral kernel} \\ [5mm]
{\large D.~Yafaev} \\ [2mm]
IRMAR, Universit\'{e} de Rennes--1, F--35325 Rennes
\end{center}
\noindent
A typical result of the paper is the following. Let ${\bf
H}_\gamma={\bf H}_0+\gamma {\bf V}$ where ${\bf H}_0$ is
multiplication by $|x|^{2l}$ and ${\bf V}$ is an integral operator
with kernel $\cos\langle x,y\rangle$ in the space $L_2({\R}^d)$.
If $l=d/2+ 2k$ for some $k= 0,1,\ldots$, then the operator ${\bf
H}_\gamma$ has infinite number of negative eigenvalues for any
coupling constant $\gamma\neq 0$. For other values of $l$, the
negative spectrum of ${\bf H}_\gamma$ is infinite for $|\gamma|>
\sigma_l$ where $\sigma_l$ is some explicit positive constant. In the
case $\pm \gamma\in (0,\sigma_l]$, the number ${\bf N}^{(\pm)}_l$ of
negative eigenvalues of ${\bf H}_\gamma$ is finite and does not
depend on $\gamma$. We calculate ${\bf N}^{(\pm)}_l$.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Variational principle for the chemical potential in the
Thomas--Fermi theory} \\ [5mm]
{\large J.~Yanez} \\ [2mm]
Facultad de Fisica, PUC, Casilia 306 -- Santiago 22, Chile
\end{center}
\noindent
We derive a new variational characterization for the chemical
potential in the Thomas--Fermi model of atoms and molecules.
We use this variational principle to obtain accurate upper bounds
for the chemical potential $\epsilon_F(N,Z)$ as a function of
the atomic number $Z$ and the number of electrons $N$. In particular
we study the behavior of $\epsilon_F(N,Z)$ for a weakly ionized
atom.
\vspace{6mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Band--gap of the spectrum in
periodically curved quantum waveguides} \\ [5mm]
{\large K. Yoshitomi} \\ [2mm]
Graduate School of Mathematics, Kyushu University \\
Fukuoka, Japan
\end{center}
\noindent
We investigate the Dirichlet Laplacian on a planar strip where the
curvature of the boundary curve is periodic with respect to the arc
length. It is proved that if the curvature is not identically 0,
there always exists some band gap of the spectrum when the strip
width is sufficiently small. We also specify the value of gap index
by using scaling argument.
\vspace{6mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Error estimates for \\ the Trotter product formula} \\
[5mm]
{\large H.~Neidhardt, \underline{V.~Zagrebnov}} \\ [2mm]
Universit\'{e} de la Mediterane\'{e}, Aix--Marseille II
\end{center}
\noindent
In addition to our talk {\em Operator-norm convergence for the
Trotter-Kato product formula} we show that under smallness
assumptions the Trotter-Kato product formula converges in
operator-norm too and, moreover, that in this case error estimates
can be found. Roughly speaking, we prove that error estimates depend
on the maximal power of the involved non-negative self-adjoint
operators, for which one power domain is embedded into another. If
this maximal power $\alpha$ obeys $1/2 < \alpha < 1$, then one has an error
estimate of order $\OO(n^{-(2\alpha-1)})$. If $\alpha = 1$, then the error
estimate is given by $\OO(\log(n)/n)$.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Construction of an exactly solvable \\ fermion model} \\
[5mm]
{\large N.~Zettili} \\ [2mm]
King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi
Arabia, and Institut de Physique, Universite de Blida, Algeria
\end{center}
\noindent
We deal here with the construction of a simple many-body
model that can be solved {\em exactly}. This model serves
as a tool for testing the validity and accuracy of
many-body approximation methods.
The model consists of a fictitious one-dimensional system
which is made of two {\em distinguishable} subsets of
fermions interacting via a schematic two-body force.
We construct the Hamiltonian of the model by means of
vector operators that are the generators of an
$SO(2,1)$ group. As these operators obey the
commutation relations of an $SO(2,1)$ Lie algebra,
they can be viewed as describing some sort of
{\em collective} vibrations; the model constructed in
terms of these operators is thus most suitable for
the study of {\em collective} motion. The Hamiltonian
depends on an adjustable parameter which regulates the
strength of the two-body interaction.
To render the size of the Hamiltonian's matrix {\em finite},
a symmetry is built into the Hamiltonian; we show that
the Hamiltonian is represented by a five-diagonal square
matrix of finite size. The energy spectrum of the model
is obtained from the diagonalization of this finite matrix.
The energy eigenvalues obtained from this diagonalization
are {\em exact}, for we don't need to resort to any
approximation in the process of diagonalization.
This model offers a rich and flexible platform for
testing various many-body approximation methods especially
those that deal with nuclear collective motion.
\vspace{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf The present state of the
study of the discrete spectrum of many--particle Hamiltonians with
homogeneous magnetic field for particles with finite masses} \\ [5mm]
{\large G.M.~Zhislin} \\ [2mm]
Radiophysical Research Institute, Nizhny Novgorod, Russia
\end{center}
\noindent
We consider the possibilities of the method of the fixation of
pseudomomentum and method of the fixation of the type of $SO(2)$
symmetry for the study of the discrete spectrum of many-particle
systems in a homogeneous magnetic field. For the first method we
formulate new results, including the spectral asymptotics for the
neutral atoms.
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
{\Large\bf Spectral analysis of the stochastic disordered Ising
model} \\ [5mm]
{\large S. Albeverio, R. Minlos, E. Scacciatelli, \underline {E.
Zhizhina}} \\ [2mm]
Moscow
\end{center}
\noindent
A tight relation between a generator of Glauber dynamics for the 1D
stochastic Ising model with random potential and a difference random
operator with off-diagonal disorder was established. We study Glauber
dynamics for the 1D Ising model with formal Hamiltonian:
$$
H(\sigma, \omega)=\,-\, \sum_{{{x,y \in Z^1}\atop {|x-y|=1 }}}
\omega_{x,y} \sigma_x \sigma_y,
$$
where
$ \omega_{x,y}= \omega_{y,x}$ are i.i.d.r.v. defined on bonds $(x,y)$,
$|x-y|=1$ of the lattice $Z^1$.
We assume that
1) the common probability distribution $p_{x,y}=p$ of the random variables
$ \omega_{x,y}$ is absolute continuous
with respect to the Lebesgue measure, and
2) the random variables $\omega_{x,y} $ are positive, finite and
bounded away from the zero:
$$
0<\gamma_1 \le \omega_{x,y} \le \gamma_2 <\infty, \quad \gamma_2=
\sup\, supp\, p.
$$
The second condition on the random field $\omega=\{\omega_{x,y} \}$
immediately implies that
for any fixed realization $\bar \omega$ of the random field and for any
$\beta$ (inverse temperature) there exists the limit Gibbs distribution
$\mu^{\beta}_{\bar \omega}$, so we can define the generator
$L_{\beta}(\bar \omega)$ of the corresponding Glauber dynamics
by the usual way. And now the main goal is to study the spectrum
of the generator -- its location and nature.
Using the combination of two techniques:
the spectral analysis of the random Schrodinger operators and
the spectral analysis of the generator for the 1D stochastic Ising model
in classical case, we have got the following result.
{\bf Theorem.}{\it The spectrum of the generator $L_{\beta}(\omega)$
is a nonrandom set with probability 1. It
is pure point with exponentially decaying eigenfunctions
(localization), and it is dense in the union of the following
segments:
$$
0, \;[-1-b,-1+b],\; [-2-2b, -2+2b],\; [-3-3b, -3+3b],....
$$
where $b=\tanh 2 \beta \gamma_2<1$ for any $\beta$.
}
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Asymptotically decreasing potentials \\ with discrete
spectra} \\ [5mm]
{\large M.~Znojil} \\ [2mm]
\'{U}stav jadern\'e fyziky AV \v{C}R, 250 68 \v{R}e\v{z}, Czech
Republic \\ {\em e-mail: znojil@ujf.cas.cz}
\end{center}
\noindent
It is well known that the central attraction $V(g,r) =
-g^2\,r^{-2}$ with a subcritical coupling $g < g_{crit} =1/2$
admits the well-behaved wavefunctions. It is also rigorously
understood that the supercritical choice of $g > g_{crit}$ would
make the particles ``fall in the origin". A possible menu of
the similar ``onset of collapse" may be much richer in
non-separable models. As an example we study a few simple
confining potentials in two dimensions. We determine their
critical couplings which become responsible for an accelerated
escape of particles in infinity in narrow tubes.
\vspace{70mm}
\noindent
{\small Last modification: June 8, 1998}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf }
\\ [5mm]
{\large } \\ [2mm]
\end{center}
\noindent
\vspace{10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%