PT microseminar:
archive 2008  2011
of the informal seminars of Doppler Institute
on quantum theory and related topics and methods,
which were taking place on the territory of the
Department of Theoretical Physics of the
Nuclear Physics Institute in Rez
during 2011, 2010, 2009 and 2008
1. the timeordered list of the talks presented during 2011:
Date and time: 
Wednesday, December 21st, 2011,
10:30 a. m.

Speaker: 
Andreas Leiser (ETH Zurich)

Title and abstract: 
Heat flows on graphs and QGraphs: Where thermodynamics meets cryptography


In 2003, Hungerbuehler and Struwe proposed a oneway function for cryptography which was based on the second law of thermodynamics. The idea was to use the heat flow which is irreversible in time. As a toy model, Hungerbuehler and Struwe illustrated the method on a finite dimensional space of bandlimited functions on the circle. Later, inspired by the problem of load balancing in computer networks, Boillat proposed to use the heat flow on Cayley graphs. We will present both ideas by several examples and conclude with some (early) ideas to use the QGraph approach to extend the HungerbuehlerStruwe toy model to more general graphs.



Date and time: 
Thursday, December 8th, 2011,
10:30 a. m.

Speaker: 
Jussi Behrndt (INM TU Graz)

Title and abstract: 
Schroedinger operators with distribution potentials


We discuss some spectral properties of selfadjoint Schr?dinger operators with delta and delta'potentials supported on smooth compact hypersurfaces.



Date and time: 
Wednesday, December 7th, 2011,
10:30 a. m.

Speaker: 
Frederik G. Scholtz (NITheP Stellenbosch)

Title and abstract: 
The entropy of dense noncommutative fermion gases


We investigate the properties of two and threedimensional noncommutative fermion gases with fixed total zcomponent of angular momentum, $J_z$, and at high density for the simplest form of noncommutativity involving constant spatial commutators. Analytic expressions for the entropy and pressure are found. The entropy exhibits nonextensive behaviour while the pressure reveals the presence of incompressibility in two, but not in three dimensions. Remarkably, for two dimensional systems close to the incompressible density, the entropy is proportional to the square root of the system size, i.e., for such systems the number of microscopic degrees of freedom is determined by the circumference, rather than the area (size) of the system. The absence of incompressibility in three dimensions, and subsequently also the absence of a scaling law for the entropy analogous to the one found in two dimensions, is attributed to the form of the noncommutativity used here, the breaking of the rotational symmetry it implies and the subsequent constraint on $J_z$, rather than the angular momentum $J$. The restoration of the rotational symmetry and the constraint of angular momentum, $J$, seem to be crucial for incompressibility in three dimensions. We briefly discuss ways in which this may be done and point out possible obstacles.



Place: 
exceptional: KM (MAFIA group), FJFI, Trojanova 13, Praha

Date and time: 
Wednesday, November 23rd, 2011,
1.30 p. m.

Speaker: 
Radek Novak (UJF AV CR & FJFI CVUT)

Title and abstract: 
The Pauli equation with nonselfadjoint boundary conditions


We present a PTsymmetric model which is a direct generalisation of the
onedimensional model introduced by Krejcirík, Bíla and Znojil [1] to the
case of multicomponent wavefunctions. We discuss the physically motivated
choice of the operator PT which now differs from the naive generalisation
from the onedimensional operator. The nonselfadjointness enters our
model through Robintype boundary conditions which are imposed on the
Pauli equation. Using the theory of sectorial forms and the First
representational theorem we %0Cfind some properties of our Hamiltonian and
conditions for its various symmetries. We further discuss several specific
examples of boundary conditions. It turns out that the spectrum of this
Hamiltonian is real for many of them, including cases which are
nonselfadjoint. This talk is based on my bachelor thesis [2].
References:
[1] D. Krejcirík, H. Bíla and M. Znojil: Closed formula for the metric in
the Hilbert space of a PTsymmetric model Journal of Physics A:
Mathematical and General, 39:1014310153, 2006.
[2] R. Novák: Pauli equation with nonHermitian PTsymmetric boundary
conditions. http://ssmf.fjfi.cvut.cz/studthes/2010/bac_thesis_novak.pdf,
2011.



Date and time: 
Friday, November 18th, 2011,
9.00 a. m.

Speaker: 
Ivan Veselic' (Chemnitz University of Technology, Saska Kamenice, Germany)

Title and abstract: 
Low lying spectrum of randomly perturbed periodic waveguides


We derive quantitative lower bounds on the lowest eigenvalue of finite segments of periodic quantum waveguides, which are perturbed randomly.
This is joint work with D. Borisov.



Date and time: 
Friday, November 18th, 2011,
8.30 a. m.

Speaker: 
Fabian Schwarzenberger (Chemnitz University of Technology, Sask? Kamenice, Germany)

Title and abstract: 
A Banach space valued ergodic theorem and the integrated density of states


In this talk we consider Cayley graphs over amenable groups. We present an ergodic theorem for Banach spacevalued, almost additive functions defined on the set of finite subsets of the vertices. We furthermore give an application of this theorem in the spectral theory of selfadjoint operators, where we get uniform convergence of the normalized eigenvalue counting functions to a limit function called integrated density of states.



Date and time: 
Wednesday, November 16th, 2011,
10.30 a. m.

Speaker: 
Tomas Ourmieres (IRMAR, Rennes)

Title and abstract: 
BornOppenheimer approximation for a conical waveguide


We will discuss the BornOppenheimer approximation for the Dirichlet Laplacian on a conical waveguide. The analysis leads to a 1D problem with nonregular potential which we use to get an estimate for the first eigenvalues of the Dirichlet Laplacian. The investigation also leads to a conjecture on the scales on which the eigenfunctions are living.



Date and time: 
Wednesday, 2. 11. 2011,
10.30 a. m.

Speaker: 
Per W. Alexandersson (Stockholm University)

Title and abstract: 
Multigraphs and sums of squares


A compact summary of some physicsmotivated algebraicgeometry results presented in paper ``Discriminants, symmetrized graph monomials, and sums of squares" written together with Boris Shapiro.
Details may be found in preprint on arXiv:1104.0589v1.



Date and time: 
27. 10. 2011,
10.30 a. m.

Speaker: 
Luca Fanelli (University of the Basque Country, Spain)

Title and abstract: 
Weak dispersion for the Schroedinger equation


We present an elementary computation which explicitly
shows some weak dispersive property of the Schroedinger equation
and some analytical applications.



Date and time: 
20. 10. 2011,
10.30 a. m.

Speaker: 
Diana Barseghyan (DI & UJF)

Title and abstract: 
Spectral estimates for a class of Schroedinger operators with infinite phase space and potential unbounded from below


We analyze twodimensional Schroedinger operators with the potential unbounded from below depending on some nonnegative parameter. We show that there is a critical value such that the spectrum of Schr\"odinger operator with parameter less than the critical value is below bounded and purely discrete, while for the parameter greater than the critical value it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.
Common work with Pavel Exner  see arXiv 1109.0168.



Date and time: 
23. 9. 2011,
10.30 a. m.

Speaker: 
Jiri Lipovsky (DI & FJFI)

Title and abstract: 
Resonances in quantum graphs


We study resonance asymptotics in the system of a scattering center with semiinfinite leads attached. First we introduce the onedimensional case  quantum graphs. By an observation by Davies and Pushnitski the coefficient by the leading term of Weyl's asymptotics can be in some cases smaller than expected. We find a general criterion for nonWeyl graphs and explain this behaviour. Furthermore, we study resonances in hedgehog manifolds (two or three dimensional Riemannian manifolds with halflines attached) and their asymptotics. In the case of a hedgehog manifold nonWeyl behaviour probably does not appear due to different dimensionality of the center and leads. We prove it in the case of one halfline and show simple examples.



Date and time: 
27. 7. 2011 (Wednesday),
10.30 a. m.

Speaker: 
Kimball A. Milton
(University of Oklahoma, Norman)

Title and abstract: 
PTSymmetric Quantum Electrodynamics


Difficulties in extending quantum electrodynamics into the
PTsymmetric but nonHermitian domain have been discussed in the
past. The Q operator, that defines the norm in the theory, and gives
the connection with the corresponding Hermitian theory, was
constructed perturbatively. However, it has yet to be proved that
this field theory possesses a unitary Smatrix. This talk
reviewed recent progress on this outstanding problem, including the
connection with the corresponding Hermitian theory. Perturbative and
nonperturbative effects were discussed.



Date and time: 
26. 5. 2011,
10.30 a. m.

Speaker: 
Jan Fischer (FzU)

Title and abstract: 
Modified perturbation expansions in QCD


Resummations and expansions using sequences of special functions of couplings in place of their usual integer powers were discussed.



Date and time: 
13. 5. 2011 (Friday),
10.30 a. m.

Speaker: 
Helena Sedivakova (FJFI & NPI)

Title and abstract: 
Effective Hamiltonian in curved quantum waveguides and when it does not work


The Dirichlet Laplacian in a curved twodimensional strip built along
a plane curve is investigated in the limit when the uniform crosssection
of the strip diminishes. We show that the Laplacian converges in a
norm resolvent sense to the well known onedimensional Schroedinger
operator whose potential is expressed solely in terms of the curvature of
the reference curve. In comparison with previous results we allow the
curves which are unbounded and whose curvature is not differentiable. On the
other hand, we present the most recent and surprising result
that there are curves with bounded curvature for which the Hamiltonian in
the waveguide is not well approximated by the conventional effective
Hamiltonian. An example of such a curve will be given.
This was a joint
work with
David Krejcirik.



Date and time: 
9. 5. 2011 (Monday),
10.30 a. m.

Speaker: 
Fabio Bagarello (Universita degli studi di Palermo)

Title and abstract: 
Nonlinear pseudobosons


After a very short introduction to pseudobosons we will show how non
selfadjoint Hamiltonians with nonequidistant real eigenvalues can be
discussed in a compact mathematical formulation.



Date and time: 
12. 4. 2011 (Tuesday),
10.30 a. m.

Speaker: 
Boris Mityagin (Columbus, Ohio, USA)

Title and abstract: 
On the convergence of perturbations of harmonic oscillator


We briefly review our recent result that the spectrum of
the perturbed harmonic oscillator
operator $L = d^2/dx^2 + x^2 + b(x)$ in $L^2(\mathbb{R}^1)$
is discrete and eventually simple
for any complex valued $L^p$function $b(x)$, $2 \leq p < \infty$ or
$L^\infty$function with the norm $\ b  L^{\infty}\ < 1$.
Its system of eigen and associated functions is an
unconditional basis in $L^2(\mathbb{R})$.
The talk was based on a joint work with James Adduci.
A more detailed account of this material is available at: arXiv:0912.2722v2 [math.SP]
(and in print).
Subjects: Spectral Theory (math.SP); Mathematical Physics (mathph)
MSC classes: 47E05, 34L40, 34L10



2010
Date and time: 
17. 12. 2010 (Friday),
10.30 a. m.

Speaker: 
Matej Tusek (KM FJFI CTU)

Title and abstract: 
Point Interaction in Momentum Representation


One, two, and three dimensional point interaction is introduced in the momentum representation. Also the twodimensional hydrogen atom with the additional onecenter point interaction is considered in the same representation. Whereas in the coordinate representation the domains of respective Hamiltonians are described by boundary conditions, in the momentum representation some integrability conditions have to be involved. Remark that in the later representation the computation of some quantities is more straightforward. The talk is based on a joint work with Pavel Stovicek.



Date and time: 
11. 10. 2010, 10.30 a. m.

Speaker: 
Gabriela Malenova (FJFI)

Title and abstract: 
Quantitative study of Hardy inequalities in twisted waveguides


We consider a planar quantum waveguide with combined Dirichlet and Neumann boundary conditions. It was shown in a 2008 paper by Kovarik and Krejcirik that a switch of the respective boundary conditions leads to an effective repulsive interaction, mathematically expressed by the existence of a Hardytype inequality for the Hamiltonian. In the talk I will present quantitative bounds on the constant appearing in the inequality obtained by variational and numerical methods.
Comment:
This is a joint work with David Krejcirik. The microseminar was followed by a drink on the occasion of a oneyear leave of absence of David Krejcirik at the Basque Center for Applied Mathematics, Bilbao.



Date and time: 
4. 10. 2010, 10.30 a. m.

Speaker: 
Helena Sedivakova (FJFI)

Title and abstract: 
Effective Hamiltonian in curved quantum waveguides as a consequence of strong resolvent convergence


The Dirichlet Laplacian in a curved twodimensional strip built along a plane curve is investigated in the limit when the uniform crosssection of the strip diminishes. We show that the Laplacian converges in a strong resolvent sense to the well known onedimensional Schroedinger operator whose potential is expressed solely in terms of the curvature of the reference curve. In comparison with previous results we allow curves which are unbounded and whose curvature is not differentiable.
This is a joint work with David Krejcirik.



Date and time: 
1. 10. 2010 (Friday), 10.30 a. m.

Speaker: 
Jakub Zelezny (FJFI)

Title and abstract: 
Closed integral formula for the metric and for the Coperator of a PTsymmetric model


This talk deals with a simple PTsymmetric nonHermitian model previously introduced by Krejcirik, Bila and Znojil. They have derived a closed formula for the metric operator which relates the problem in the Hilbert space to a Hermitian one. We present a new, integral form of this metric operator, and use it to perform a perturbative computation of the corresponding Hermitian Hamiltonian. We also present an integral formula for a different metric operator, which is associated with the Csymmetry operator.



Date and time: 
13. 5. 2010, 10.30 a. m.

Speaker: 
Sergii Kuzhel (Institute of Mathematics NAS, Kiev, Ukraine)

Title and abstract: 
On JSelfAdjoint Operators with CSymmetries: Extension Theory Approach


The development of PTsymmetric quantum mechanics achieved during the last decade gives rise to a lot of new useful notions and interesting mathematical problems in the theory of Jselfadjoint operators and, more generally, in the Krein spaces theory.
The aim of the report is to present some problems arising in this field. The main attention will be paid to the theory of Jselfadjoint operators with Csymmetry in the extension theory framework. We will discuss the recent notion of stable Csymmetry for Jselfadjoint extensions of a symmetric operator S. The spectral analysis of the Jselfadjoint extensions of S inside the sector of stable Csymmetry and its boundary will be carried out.



Date and time: 
29. 4. 2010, 10.30 a. m.

Speaker: 
Adam Smetana (NPI)

Title and abstract: 
Origin of Mass from Flavor Gauge Dynamics


We discuss implications of the recently suggested model of electroweak symmetry breaking (EWSB) in which the apparent flavor symmetry of particle dynamics is gauged. To trigger the EWSB the flavor gauge symmetry is spontaneously broken. The anomaly freedom requires to introduce righthanded neutrinos in various flavor representations. Their number is limited not to disturb the asymptotic freedom. The dynamics generates both Dirac and Majorana mass matrices for neutrinos. Consequently, they break a `sterile' symmetry of the righthanded neutrino sector giving rise to a to rich spectrum of majorons. The majorons couple to the sterile neutrinos, and by loops also to a visible matter. Both sterile neutrinos, and majorons can possibly account for the observed dark matter component of the Universe.



Date and time: 
11. 3. 2010, 10.30 a. m.

Speaker: 
Takuya Mine (Kyoto Institute of Technology & CTU in Prague)

Title and abstract: 
Selfadjoint extensions of the Schroedinger operators with singular magnetic fields


It is wellknown that all the selfadjoint extensions of a differential operator with singular coefficients with the minimal operator domain can be determined by the Krein  von Neumann theory. However, we cannot apply this theory when we don't know explicitly the deficiency subspaces of the operator, e.g., the twodimensional Schroedinger operator with two or more Aharonov  Bohm magnetic fields. In this talk, we give a simple method to determine the selfadjoint extensions in the case we know only the local structure of the deficiency subspaces.



Date and time: 
7. 1. 2010, 10.30 a. m.

Speaker: 
Petr Sulc (FJFI & Los Alamos Nat. Lab.)

Title and abstract: 
Belief propagation and its application to graph partitioning


We study the problem of graph partitioning, which is equivalent to
the groundstate problem in the ferromagnetic Ising model at a fixed
magnetization. This offers a typical situation of the applicability of the
methods of statistical mechanics and message passing algorithms. We use
BethePeierls approximation (i.e. belief propagation algorithm) whose broad
spectrum of applications (e.g. spin glasses, SAT problems, graph coloring)
attracts a lot of interest.
In this context, we show how to incorporate the constraint of fixed
magnetization into the belief propagation equations. This approach provides
an efficient heuristic solver of the graph partitioning problem and we will
review briefly possible applications.
This is a joint work with Lenka Zdeborova (viz
http://arxiv.org/abs/0912.3563
).



2009
Date and time: 
3. 12. 2009, 10.30 a. m.

Speaker: 
Petr Siegl (NPI & DI)

Title and abstract: 
PTsymmetric systems on curved manifolds II


We introduce nonHermitian models in curved twodimensional manifolds based on striplike geometries with PTsymmetric boundary conditions. We present both analytic and numerical results on the study of interplay between geometry and spectrum. Basic properties of the operators are proved by perturbative methods.
We particularly investigate models defined on manifolds of the constant curvature and subject to PTsymmetric Robin boundary conditions. We describe spectral effects caused by the curvature. This is a joint work with David Krejcirik.



Date and time: 
26. 11. 2009, 10.30 a. m.

Speaker: 
Milos Tater (NPI & DI)

Title and abstract: 
Electrostatic models for zeros of polynomials


We review some electrostatic models (electrostatic equilibrium problems)
that are linked to zero distribution of polynomial solutions of hypergeometric and Heun equations. We discuss Jacobi and Freud polynomials, and Chebyshev rational function in detail.



Date and time: 
18. 11. 2009, 10.30 a. m.

Speaker: 
Vit Jakubsky (NPI & DI)

Title and abstract: 
Superconformal symmetry of spinless AharonovBohm system


We find a hidden supersymmetry in the spinless AharonovBohm system. Similar superalgebraic structure is observed in two different selfadjoint extensions of the formal AharonovBohm Hamiltonian. We show that the hidden supersymmetry is compatible with the conformal invariance of the system and we find an additional nonlocal dynamical symmetry.
Source: arXiv:0906.4055



Date and time: 
16. 11. 2009, 10.30 a. m.

Speaker: 
Hynek Bila (NPI & DI)

Title and abstract: 
New metric for a model with point interaction


New solution for the metric valid for a nonHermitian 1D model of P. Siegl will be presented. The solution, itself timeindependent, arises from calculations where timedependent Hamiltonian is concerned



Date and time: 
11. 11. 2009, 11.15 a. m.

Speaker: 
Petr Siegl (NPI & DI)

Title and abstract: 
PTsymmetric models in curved manifolds


Preliminary version of the 20 min. talk, to be presented during
``Doktorand`s days" (13. 11. 2009, 14.00, FJFI Prague, Trojanova st.).
Work in progress, results of collaboration with David Krejcirik.



Date and time: 
25. 6. 2009, 10.30 a. m.

Speaker: 
Takuya Mine
(Kyoto Institute of Technology &
The Czech Technical University in Prague)

Title and abstract: 
Absolutely continuous spectrum for the Schroedinger operators with periodic mangetic fields


It is wellknown that the Schroedinger operators with periodic scalar potentials have the purely absolutely continuous spectrum in general. But it is not the case for the Schroedinger operators with periodic magnetic fields. In this talk, we give some examples of (possibly singular) magnetic fields, with which the Schroedinger operators have the absolutely continuous spectrum, at least in a part of the spectrum.



Date and time: 
18. 6. 2009, 10.30 a. m.

Speaker: 
Aldo Pratelli (Dip. Mat., Univ. Pavia)

Title and abstract: 
Stability versions for isoperimetric inequality and other functional inequalities


We will shortly describe the known results about the stability of some geometric and functional inequalities: the main example is the isoperimetric inequality, but the results are valid even for other inequalities, such as the Sobolev inequalities, the isodiametric inequality, and other inequalities of this kind. The study of such versions is quite old but there are many new results obtained via a new technique.



Date and time: 
11. 6. 2009, 10.30 a. m.

Speaker: 
Hynek Bila (NPI & DI)

Title and abstract: 
Timedependent metric for a quantummechanical model with point interaction


Discussion of the problem of determination of the evolution operator and of the timedependent metric operators for the Siegel `s 1D quantummechanical model with point interaction



Date and time: 
20. 1. 2009 (irregular: Tuesday), 10.30 a. m.
(reclassified as the Institute's seminar  lasted cca 75 minutes)

Speaker: 
George Japaridze (Center for Theoretical Studies of Physical Systems,
Clark Atlanta University
Atlanta, GA, USA)

Title and abstract: 
IceCube  the next generation neutrino telescope at the South Pole


IceCube is a large neutrino telescope of the next generation being
constructed in the Antarctic Ice Sheet near the South Pole.
The main goal of the IceCube project is to extend the region
of the Universe explored by neutrinos, and to learn about the origin
of the highest energy cosmic rays.
I present the science case, conceptual design and the status of the
detector, and the science results anticipated.



2008
Date and time: 
6. 11. 2008, 10.30 a. m.

Speaker: 
Pedro Freitas (University of Lisbon)

Title and abstract: 
Will it burst?
or
Everything you don't want to know about brain aneurysms.


We consider a model that has been proposed by Shah and Humphrey for the dynamical behaviour of brain aneurysms. We prove that for constant blood pressure the model does behave as expected, and there is convergence to equilibrium. However, for periodic forcing terms (pressure), the outcome might be different.



Date and time: 
2. 10. 2008, 10.30 a. m.
(reclassified as the Institute's seminar  took cca 45 minutes)

Speaker: 
Daniel Hook (Imperial College, London)

Title and abstract: 
Isospectral Hamiltonians


I will review techniques developed by Andrianov, Bender, Brody, Chen, Jones, Mateo, Milton, Ogilvie and Rivers for constructing corresponding PTSymmetric and Hermitean Hamiltonians. We will see how this methodology may be extended to create an infinite tower of pairs of isospectral Hamiltonians. We find that, in general, the pairs are composed of PTSymmetric Hamiltonians and that the PTSymmetry/Hermitean correspondence is a special case. This talk will focus on the numerical aspects of these calculations and will be accessible to a nonspecialist audience.



Date and time: 
18. 9. 2008, 10.30 a. m.

Speaker: 
Olaf Post (TU Berlin)

Title and abstract: 
Approximation of delta and delta'interactions by Laplaceans on manifolds


We show that there exists a family of manifolds together with a Schroedingeroperator on it such that the manifold is close to a metric graph and the Schroedingeroperator converges to the Schroedingeroperator on the graph with delta and delta'interactions at a vertex. I will roughly explain the main ideas.
==============
During the time for questions the discussion also involved the eigenvalue bracketing presented during the Tuesday's DI ``Quantum circle" seminar based on the arXiv: 0804.1076 paper with F. Lledo where the authors say that they
developed eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the ``exceptional'' values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps (comments on the arXiv: 0804.1076 paper with F. Lledo).



Date and time: 
7. 8. 2008, 10.30 a. m.

Speaker: 
Jaroslav Dittrich (NPI & DI)

Title and abstract: 
On the connectedness notion


This is a short undergraduate level lecture for our education, not a seminar on original results. The definitions of the connected and pathconnected sets will be reminded. An example of a set in the Euclidean plane which is connected but pathdisconnected will be shown.



Date and time: 
June 27th (postponed from June 5th and
presented as a part of the seminar of Theory Group)

Speaker: 
Stefan Gmuca (FZU SAV Bratislava)

Title and abstract: 
Relativistic mean field model: Fock term


A relativistic densitydependent mean field approach for nuclear matter is presented, where the coupling constants are obtained from the relativistic HartreeFock model by mapping the Fock terms onto the relativistic mean field Lagrangian. The obtained results are compared with the DiracBrueckner HartreeFock calculations of nuclear matter.



Date and time: 
12. 6. 2008, time shifted to 14.00

Speaker: 
Bertha CuadrosMelgar
(Departamento de Fisica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile & DI)

Title and abstract: 
Brane World Cosmology Models


We will introduce the idea of the universe as a membrane and revise the known cosmological models in this context and its implications for brane observers.



Date and time: 
5. 6. 2008, 10.30 a. m.

Speaker: 
Raffaele Carlone (NPI & DI)

Title and abstract: 
Spectral properties of spin dependent point interactions


We will analyze a completely solvable quantum system where formation and time decay properties of a metastable state can be explicitly investigated.



Date and time: 
29 5. 2008, 10.30 a. m.

Speaker: 
Jaroslav Dittrich (NPI & DI)

Title and abstract: 
Integrals of motion for quantum spin chains with elliptic Weierstrass function interaction


We prove the commutativity and linear independence of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments. Common work with V. I. Inozemtsev (Regular and Chaotic Dynamics 19 (2008), 1926).



Date and time: 
22 5. 2008, 10.30 a. m.

Speaker: 
Angela Mestre (DI & KM FJFI)

Title and abstract: 
The combinatorics of npoint functions via Hopf algebra in quantum field theory


We use the Hopf algebra structure of the timeordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected npoint functions. Moreover, we derive simple relations between complete, connected and 1particle irreducible npoint functions. Compared to traditional functional methods our approach is much more intrinsic and leads to efficient algorithms suitable for concrete computations.



Date and time: 
12. 5. 2008, 10.30 a. m.

Speaker: 
Stefan RauchWojciechowski (Linkoeping University)

Title and abstract: 
Schroedinger, Newton, KdV and Harry Dym equations


The energy dependent Schroedinger spectral problem and integrable systems of Newton equations embedded into hierarchies of KdV and Harry Dym soliton equations will be discussed.



Date and time: 
28 4. 2008, 10.30 a. m.

Speaker: 
L'ubomira Balkova (DI & KM FJFI)

Title and abstract: 
Discrete Schrodinger operators with aperiodic potentials


This talk will be about discrete Schroedinger operators with potentials generated by a primitive substitution. We are particularly interested in substitutions associated with the socalled Parry numbers. Putting together several methods relying on combinatorial properties of fixed points of Parry substitutions, we deduce for which Parry numbers, the corresponding Schroedinger operator has purely singular continuous spectrum. In such cases, its potential is believed to be a suitable model of the potential in a quasicrystalline material.

Material available: 
In advance, transparencies are available on the following subjects:
1 1D Schroedinger equation and Schroedinger operator
2 Discrete Schroedinger operator
3 Potentials generated by primitive substitutions
4 Known spectral properties
5 Exclusion of pure point spectrum
6 Combinatorial methods
7 Example: Simple Parry substitutions
8 Spectra of Schroedinger operators associated with simple Parry substitutions

Date and time: 
24 4. 2008, 10.30 a. m.

Speaker: 
Hynek Bila (NPI & DI)

Title and abstract: 
On PseudoHermitian Hamiltonians I


The first part of a series of microlectures on the recent results on pseudoHermiticity. Definitions and notation and the abstracttype review of the contents of the PhD thesis.

Material available: 
In electronic form, upon
emailed
request.

Date and time: 
17 4. 2008, 10.30 a. m.

Speaker: 
Milos Tater (NPI & DI)

Title and abstract: 
Spectral methods in application to quantum waveguides


Three different computational methods for determining the eigenvalues and eigenfunctions of the Schrodinger equation are discussed. The methods are based on solutions of timedependent Schrodinger equation and the study of (auto)correlation function. A possible application to bound states of quantum waveguides is presented.

Date and time: 
10 4. 2008, 10.30 a. m.

Speaker: 
Vit Jakubsky (Departamento de Fisica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile)

Title and abstract: 
Nonlinear symmetry where you wouldn't expect it


Symmetries of physical systems reflect peculiar dynamics of the considered models. We will discuss simple quantum systems with nonlinear symmetries which underlie their specific spectral properties.
We reveal an unexpected hidden supersymmetric structure in a selfisospectral system constructed on the base of the periodic finitegap associated Lame equation. It admits three different choices of the Z_2grading, under which three basic nontrivial integrals of motion coherently change their fermionic/bosonic nature, and generate a certain nonlinear supersymmetry. These integrals reflect the band structure, its separability, and characteristic properties of the bandedge states of the system, which can be interpreted as an electron in onedimensional crystal produced by periodic electric and magnetic fields.

Material available: 
arXiv:0801.1671v2 [hepth] 16 Jan 2008.

Date and time: 
26. 3. 2008, 10.30 a. m.

Speaker: 
David Krejcirik (NPI & DI)

Title and abstract: 
NonHermitian spectral effects in a PTsymmetric waveguide


The ultimate objective of this talk is to present a nice animation obtained recently in collaboration with Milos Tater.

Date and time: 
20. 3. 2008, 10.30 a. m.

Speaker: 
Jan Kubieniec (CTU & DI & NPI)

Title and abstract: 
A method for calculating the spectrum of the Casimir operator of the group SO(1,2) in the supplementary series of unitary representations of the group SO(1,3)


Experiments show that electrically charged matter has charge being an integer multiple of a certain elementary charge. Quantum electrodynamics does not explain this quantization and the charge is only a constant parameter in this theory. The proposal of an explanation of this quantization is the socalled quantum mechanics of the electric charge constructed by A. Staruszkiewicz, Ann. Phys. (N.Y.) 190, 354 (1989). Within the framework of Staruszkiewicz's theory there exists a functional relation between the elementary charge and the parameter which labels unitary representations of the group SO(1,3) from the supplementary series. To calculate the elementary charge it is thus necessary to select a single representation from the supplementary series. This is a motivation to investigate the structure of the supplementary series. One of approaches is an analysis of the form of the spectrum of the Casimir operator of the group SO(1,2). I shall discuss a method for calculating the spectrum of this operator.

Date and time: 
13. 3. 2008, 10.30 a. m.

Speaker: 
Jan Novotny (NPI)

Title and abstract: 
Tunable quantumtoboggan paths


The recent article, "Quantum Toboggans with Two Branch Points" (M. Znojil, Phys. Lett. A 372, Issue 5, 28 January 2008, Pages 584590, http://dx.doi.org/10.1016/j.physleta.2007.07.072, arXiv:0708.0087v1 [quantph] 1 Aug 2007), caught the attention of one of the Wolfram Research agents who suggested that it could be developed into an interesting Demonstration to add to The Wolfram Demonstrations Project http://demonstrations.wolfram.com. The preliminary form of such a demonstration submission will be presented. If accepted to The Wolfram Demonstrations Project website, it might expect attracting over 50,000 hits a week in the average.

Date and time: 
6. 3. 2008, 10.30 a. m.

Speaker: 
JunHua Chen (FJFI & DI)

Title and abstract: 
Equivalence of a Complex PT Symmetric Quartic Hamiltonian and
a Hermitian Quartic Hamiltonian with an Anomaly


In 1993, Buslaev and Grecchi
(and, independently,
Jones and Mateo in 2006) revealed that the
fourthpower potential with the negative coupling strength
leads to the same spectrum as
another fourthpower potential with the positive coupling strength
and with an anomalous, linearly rising term.
We shall explain and discuss this coincidence of the spectra
using two alternative methods.

Material available: 
[1] J.H. Chen,
CLASSICAL AND QUANTUM PT SYMMETRIC THEORIES (PhD thesis, Washington University, March 2007).
[2] C. M. Bender, D. C. Brody, J.H. Chen, H. F. Jones, K. A. Milton, and M. C.
Ogilvie, Phys. Rev. D 74, 025016
(2006), arXiv:hepth/0605066.

Date and time: 
28. 2. 2008, 10.30 a. m.

Speaker: 
JunHua Chen (FJFI & DI)

Title and abstract: 
PTsymmetric interpretation of the Lee model


In the fifties, T. D. Lee
introduced a schematic though fairly popular model
of interacting
``proton", ``neutron" and ``pion" (without antiparticles)
in which mass, wave functions and charge can be renormalized in closed
form.
Unfortunately, until recently, the theory has been considered inconsistent
because it seemed to predict the existence of a ``ghost" state
exhibiting a negative square of the norm.
We revealed that
once one introduces a suitable metric, the ghosts disappear and
the whole quantum model acquires the entirely
standard physical meaning.
The technical core of this innovative reinterpretation of the model
will be briefly summarized.

Material available: 
PhD thesis and also
C. M. Bender, S. F. Brandt, J.H. Chen, and Q. Wang,
Ghost Busting: PT 
Symmetric Interpretation of the Lee Model, Phys. Rev. D 71, 025014 (2005),
arXiv:hepth/0411064.

Date and time: 
21. 2. 2008, 10.30 a. m.

Speaker: 
Hynek Bila (NPI & DI)

Title and abstract: 
Time dependent nonHermitian Hamiltonians


The problems arising with explicit time dependence introduced into an
evolution driven by pseudoHermitian Hamiltonian will be discussed. Two
alternative solutions are compared on the example of a simple matrix
model.

Material available: 
incorporated in PhD thesis.

Date and time: 
14. 2. 2008, 10.30 a. m.

Speaker: 
Miloslav Znojil (NPI & DI)

Title and abstract: 
Quantum knots


It is known that besides the usual unitary mappings $\Omega = 1/\Omega^\dagger$ between the equivalent representations of the physical Hilbert space of Quantum Mechanics (often, Fourier transformations), the generalized nonunitary maps $\Omega \neq 1/\Omega^\dagger$ can also help to simplify the analysis. We adapt the standard Dirac's notation and recollect the Buslaev's and Grecchi's repulsive quartic oscillator Hamiltonian as an example. Then we propose the whole new class of the models of the similar type, characterized by a complexification of the path C of the (obviously, not observable!) ``coordinates". An exactly solvable potentialless Schroedinger equation is finally chosen for illustration. In it, the dynamical (i.e., in our example, confining) role of the traditional potentials V(x) is shown to be taken over by the mere topologically nontrivial shape of C. Our construction evokes several new open questions in physics (PTsymmetric wave packets at a single energy?) as well as in mathematics (a threeHilbertspace generalized formulation of Quantum Mechanics?).

Material available: 
arXiv: 0802.1318v1 [quantph] 10 Feb 2008, published in Phys. Lett. A 372 (2008) 35913596.

Date and time: 
7. 1. 2008 (irregular), 10.30 a. m.

Speaker: 
Petr Siegl (DI & Paris VII & FJFI)

Title and abstract: 
QuasiHermitian models I


An introductory, unprepared report on early stages of diploma work: definitions, projects and some preliminary ideas.

Material available: 
diploma thesis (in preparation).

Reminder:
the list of subjects included
 analytic and algebraic methods
 constructive Quantum Mechanics
 exactly solvable problems
 perturbation expansions
 nonHermitian quantum models
 computer tricks
 relativistic equations
 and so on.

the secretary of the microseminar was 
Miloslav Znojil

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