PT microseminar (archive 2008)

Informal seminar of Doppler Institute
on quantum theory and related topics and methods,
taking place, usually, on the territory of the
Department of Theoretical Physics of the
Nuclear Physics Institute in Rez

The complete list of the talks during 2008

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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