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 time-ordered 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 one-way 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 band-limited 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 Hungerbuehler-Struwe 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 self-adjoint 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 non-commutative fermion gases
We investigate the properties of two- and three-dimensional non-commutative fermion gases with fixed total z-component 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 non-extensive 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 non-commutativity 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 non-selfadjoint boundary conditions
We present a PT-symmetric model which is a direct generalisation of the one-dimensional 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 one-dimensional operator. The non-selfadjointness enters our model through Robin-type 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 non-selfadjoint. This talk is based on my bachelor thesis [2].
[1] D. Krejcirík, H. Bíla and M. Znojil: Closed formula for the metric in the Hilbert space of a PT-symmetric model Journal of Physics A: Mathematical and General, 39:10143-10153, 2006.
[2] R. Novák: Pauli equation with non-Hermitian PT-symmetric boundary conditions., 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 space-valued, 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: Born-Oppenheimer approximation for a conical waveguide
We will discuss the Born-Oppenheimer approximation for the Dirichlet Laplacian on a conical waveguide. The analysis leads to a 1-D problem with non-regular 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 physics-motivated algebraic-geometry 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 two-dimensional 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 one-dimensional 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 non-Weyl 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 non-Weyl 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: PT-Symmetric Quantum Electrodynamics
Difficulties in extending quantum electrodynamics into the PT-symmetric but non-Hermitian 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 S-matrix. 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 two-dimensional strip built along a plane curve is investigated in the limit when the uniform cross-section of the strip diminishes. We show that the Laplacian converges in a norm resolvent sense to the well known one-dimensional 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: Non-linear pseudo-bosons
After a very short introduction to pseudo-bosons we will show how non self-adjoint Hamiltonians with non-equidistant 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 (math-ph)
MSC classes: 47E05, 34L40, 34L10


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 two-dimensional hydrogen atom with the additional one-center 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 Hardy-type 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 one-year 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 two-dimensional strip built along a plane curve is investigated in the limit when the uniform cross-section of the strip diminishes. We show that the Laplacian converges in a strong resolvent sense to the well known one-dimensional 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 C-operator of a PT-symmetric model
This talk deals with a simple PT-symmetric non-Hermitian 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 C-symmetry operator.

Date and time: 13. 5. 2010, 10.30 a. m.
Speaker: Sergii Kuzhel (Institute of Mathematics NAS, Kiev, Ukraine)
Title and abstract: On J-Self-Adjoint Operators with C-Symmetries: Extension Theory Approach
The development of PT-symmetric quantum mechanics achieved during the last decade gives rise to a lot of new useful notions and interesting mathematical problems in the theory of J-self-adjoint 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 J-self-adjoint operators with C-symmetry in the extension theory framework. We will discuss the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The spectral analysis of the J-self-adjoint extensions of S inside the sector of stable C-symmetry 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 right-handed 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 right-handed 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: Self-adjoint extensions of the Schroedinger operators with singular magnetic fields
It is well-known that all the self-adjoint 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 two-dimensional Schroedinger operator with two or more Aharonov - Bohm magnetic fields. In this talk, we give a simple method to determine the self-adjoint 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 ground-state 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 Bethe-Peierls 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 ).


Date and time: 3. 12. 2009, 10.30 a. m.
Speaker: Petr Siegl (NPI & DI)
Title and abstract: PT-symmetric systems on curved manifolds II
We introduce non-Hermitian models in curved two-dimensional manifolds based on strip-like geometries with PT-symmetric 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 PT-symmetric 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 Aharonov-Bohm system
We find a hidden supersymmetry in the spinless Aharonov-Bohm system. Similar superalgebraic structure is observed in two different self-adjoint extensions of the formal Aharonov-Bohm Hamiltonian. We show that the hidden supersymmetry is compatible with the conformal invariance of the system and we find an additional non-local 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 non-Hermitian 1D model of P. Siegl will be presented. The solution, itself time-independent, arises from calculations where time-dependent Hamiltonian is concerned

Date and time: 11. 11. 2009, 11.15 a. m.
Speaker: Petr Siegl (NPI & DI)
Title and abstract: PT-symmetric 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 well-known 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: Time-dependent metric for a quantum-mechanical model with point interaction
Discussion of the problem of determination of the evolution operator and of the time-dependent metric operators for the Siegel `s 1D quantum-mechanical 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.


Date and time: 6. 11. 2008, 10.30 a. m.
Speaker: Pedro Freitas (University of Lisbon)
Title and abstract: Will it burst?
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 PT-Symmetric 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 PT-Symmetric Hamiltonians and that the PT-Symmetry/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 Schroedinger-operator on it such that the manifold is close to a metric graph and the Schroedinger-operator converges to the Schroedinger-operator 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 path-connected sets will be reminded. An example of a set in the Euclidean plane which is connected but path-disconnected 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 density-dependent mean field approach for nuclear matter is presented, where the coupling constants are obtained from the relativistic Hartree-Fock model by mapping the Fock terms onto the relativistic mean field Lagrangian. The obtained results are compared with the Dirac-Brueckner- Hartree-Fock calculations of nuclear matter.

Date and time: 12. 6. 2008, time shifted to 14.00
Speaker: Bertha Cuadros-Melgar (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), 19-26).

Date and time: 22 5. 2008, 10.30 a. m.
Speaker: Angela Mestre (DI & KM FJFI)
Title and abstract: The combinatorics of n-point functions via Hopf algebra in quantum field theory
We use the Hopf algebra structure of the time-ordered 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 n-point functions. Moreover, we derive simple relations between complete, connected and 1-particle irreducible n-point 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 Rauch-Wojciechowski (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 so-called 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 1-D 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 Pseudo-Hermitian Hamiltonians I
The first part of a series of microlectures on the recent results on pseudo-Hermiticity. Definitions and notation and the abstract-type 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 time-dependent 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 non-linear symmetries which underlie their specific spectral properties. We reveal an unexpected hidden supersymmetric structure in a self-isospectral system constructed on the base of the periodic finite-gap associated Lame equation. It admits three different choices of the Z_2-grading, 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 band-edge states of the system, which can be interpreted as an electron in one-dimensional crystal produced by periodic electric and magnetic fields.
Material available:
arXiv:0801.1671v2 [hep-th] 16 Jan 2008.

Date and time: 26. 3. 2008, 10.30 a. m.
Speaker: David Krejcirik (NPI & DI)
Title and abstract: Non-Hermitian spectral effects in a PT-symmetric 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 so-called 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 quantum-toboggan paths
The recent article, "Quantum Toboggans with Two Branch Points" (M. Znojil, Phys. Lett. A 372, Issue 5, 28 January 2008, Pages 584-590,, arXiv:0708.0087v1 [quant-ph] 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 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: Jun-Hua 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 fourth-power potential with the negative coupling strength leads to the same spectrum as another fourth-power 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,
(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),

Date and time: 28. 2. 2008, 10.30 a. m.
Speaker: Jun-Hua Chen (FJFI & DI)
Title and abstract: PT-symmetric 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:hep-th/0411064.

Date and time: 21. 2. 2008, 10.30 a. m.
Speaker: Hynek Bila (NPI & DI)
Title and abstract: Time dependent non-Hermitian Hamiltonians
The problems arising with explicit time dependence introduced into an evolution driven by pseudo-Hermitian 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 non-unitary 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 (PT-symmetric wave packets at a single energy?) as well as in mathematics (a three-Hilbert-space generalized formulation of Quantum Mechanics?).
Material available:
arXiv: 0802.1318v1 [quant-ph] 10 Feb 2008, published in Phys. Lett. A 372 (2008) 3591-3596.

Date and time: 7. 1. 2008 (irregular), 10.30 a. m.
Speaker: Petr Siegl (DI & Paris VII & FJFI)
Title and abstract: Quasi-Hermitian models I
An introductory, unprepared report on early stages of diploma work: definitions, projects and some preliminary ideas.
Material available:
diploma thesis (in preparation).


the list of subjects included
  • analytic and algebraic methods
  • constructive Quantum Mechanics
  • exactly solvable problems
  • perturbation expansions
  • non-Hermitian quantum models
  • computer tricks
  • relativistic equations
  • and so on.

the secretary of the microseminar was Miloslav Znojil

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