PT microseminar (archive 2007)

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 2007

Date and time: 11. 12. 2007, 2.15 p. m.
Speaker: Jan Vrbik (Brock University, Canada)
Title and abstract: Monte Carlo computations of ground-state-energy derivatives
In the Monte-Carlo calculations of the ground-state energies E, say, of small molecules, the statistical errors of results pose a major obstacle when one tries to deduce the molecule's geometry, vibrational frequencies and polarizabilities since these quantities are defined in terms of the derivatives of E with respect to a parameter. We shall describe a specific method suitable for such a purpose. In essence, a low-cost extension of the Monte Carlo algorithm is obtained via an appropriate ad hoc incorporation of certain correlations.
Material available:
in its published version in International Journal of Quantum Chemistry, Vol 108, 493502 (2008).

Date and time: 25. 10. 2007, 10.30 a. m.
Speaker: Milos Tater (NPI & DI)
Title and abstract: Spectra of a PT-symmetric wave guide
We summarize results of paper by Denisov, Krejcirik. Then we show how to treat the bound states numerically. Finally, we show that the numerically obtained spectra are surprisingly rich and stimulate further investigation.
not available yet (pdf) or (zipped)

Date and time: 18. 10. 2007, 10.30 a. m.
Speaker: Eva-Maria Graefe (University of Kaiserslautern)
Title and abstract: The spectrum of an open two-mode Bose-Hubbard system
In a two-mode approximation a Bose-Einstein condensate in a double-well trap can be described by a many particle Hamiltonian of Bose-Hubbard type. We introduce effective complex energies to the modes to describe a coupling to a continuum and obtain a non-Hermitian matrix model. In dependence on the systems parameters the eigenvalues show intricate structures of avoided and real crossings, as well as bifurcations. In the present talk the effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed.
not available yet (pdf) or (zipped)

Date and time: 11. 10. 2007, 10.30 a. m.
Speaker: Denis Kochan (Comenius University, Bratislava)
Title and abstract: Geometry of the Berry Phase
In 20 minutes I will try to elucidate what type of geometrical data (spectral line bundle and Berry-Barry connection) are connected with adiabatic evolution in QM. Unfortunatelly, no new results will be presented here.
available here (pdf) or here (zipped)

Date and time: 1. 10. 2007 (Monday), 10.30 a. m.
Speaker: R. Roychoudhury (Indian Statistical Institute, Kolkata)
Title and abstract: A few remarks on pseudo-Hermitian Hamiltonians
Brief review of results on PT symmetric and supersymmetric quantum mechanical models will concentrate on a buildup of their perturbative connection to position-dependent-mass Hamiltonians and on the constructions of the C operators.

Date and time: 19. 9. 2007 (Wednesday), 10.30 a. m.
Speaker: Martin Jilek (FJFI)
Title and abstract: Straight quantum waveguide with Robin boundary conditions
We investigate spectral properties of a quantum particle confined to an infinite straight planar strip by imposing Robin boundary conditions with variable coupling. Assuming that the coupling function vanishes at infinity, we localise the essential spectrum and derive a sufficient condition which guarantees the existence of bound states. Further properties of the associated eigenvalues and eigenfunctions are studied numerically by the mode-matching technique.

Date and time: 28. 6. 2007, 10.30 a. m.
Speaker: Shogo Tanimura (Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Japan)
Title and abstract: A new formulation of the uncertainty relation and its implication for the double-slit experiment
It is well-known that Heisenberg formulated the uncertainty relation in 1927. He induced it via analyses of specific thought experiments. Kennard (1927) proved the uncertainty relation in a general manner and Robertson (1929) generalized it for any non-commuting observables. Recently, Ozawa (2003) re-examined the physical implication of the uncertainty relation and formulated a new type of the uncertainty relation. I examined a physical implication of the Young double-slit experiment to compare the roles of the Kennard-Robertson uncertainty relation and of the Ozawa uncertainty relation. I also proposed a possible experimental scheme to test the uncertainty relation between interference visibility and path distinguishability. I would like to report a brief summary of these works. An e-print is available:

Date and time: 17. 5. 2007, 10.30 a. m.
Speaker: Tamas Fulop (DI & NPI)
Title and abstract: Physical role attributed to self-adjointness in quantum mechanics
A collection of pros and cons of keeping self-adjointness and the pros and cons of loosening this restriction, from the point of view of physical, quantum statistical, interpretation. The aim is to pose the questions that have to be addressed for, say, a PT-symmetric quantum mechanics. We plan to show the aspects from which self-adjointness seems to be too tight for physics.

Date and time: 3. 5. 2007, 10.30 a. m.
Speaker: Hynek Bila (DI & NPI)
Title and abstract: A few remarks on the Shin's proofs
A brief review of some results by Kwang Shin. References:
1. On the Eigenproblems of PT-Symmetric Oscillators. Journal of Mathematical Physics, 42 (6): 2513-2530, 2001. ArXiv: math-ph/0007006
2. On the Reality of the Eigenvalues for a Class of PT-Symmetric Oscillators. Communications in Mathematical Physics, 229 (3): 543-564, 2002. ArXiv: math-ph/0201013
3. On half-line spectra for a class of non-self-adjoint Hill operators. Mathematische Nachrichten, 261-262: 171-175, 2003. ArXiv: math-ph/0308032
4. Eigenvalues of PT-symmetric oscillators with polynomial potentials. Journal of Physics A: Mathematical and General, 38: 6147-6166, 2005. ArXiv: math.SP/0407018
5. The potential (iz)m generates real eigenvalues only, under symmetric rapid decay conditions. Journal of Mathematical Physics, 46: 082110 (17 pages), 2005. ArXiv: hep-ph/0207251

Date and time: 5. 4. 2007, 10.30 a. m.
Speaker: Jan Kubieniec (CTU & DI & NPI)
Title and abstract: On the representations of Lorentz group
A brief review of some new results. References:
1. J. Kubieniec, On the structure of the supplementary series of unitary, irreducible representations of the proper, orthochronous Lorentz group, J. Math. Phys. 46, 103518 (2005)
2. J. Kubieniec, On unitary, irreducible representations of the proper, orthochronous Lorentz group, J. Math. Phys. 46, 053502 (2005)

Date and time: 29. 3. 2007, 11:00 a. m.
Speaker: Vit Jakubsky (NPI & DI)
Title and abstract: Project Chile
Quantization of a particle on a hyperboloid. A brief comment added on the broader research plans for the nearest future. Discussion.

Date and time: 8. 3. 2007, 10.30 a.m.
Speaker: Jaroslav Dittrich (NPI & DI)
Title and abstract: Derivative in L^2-norm
The derivative of a function exists in the norm of L^2(R) space if and only if the function belongs to the Sobolev space W^{1,2}(R) and then it equals to the classical derivative almost everywhere.

Date and time: unusual: Tuesday, February 20th, 2007, 9.30 a.m.
Place: unusual: Villa Lanna in Prague
Speakers: eight speakers: see the separate webpage

Date and time: 25. 1. 2007, 10.30 a.m.
Speaker: Denis I. Borisov [NPI (Marie Curie Fellow) & DI (visitor)& Pedagogical University, Ufa (on leave)]
Title and abstract: A PT-symmetric waveguide
We consider a planar waveguide of constant width with PT-symmetric Robin boundary conditions and study the spectrum of this system. We focus our attention on the case when the coefficient in the boundary condition is perturbed by a compactly supported function. We prove that the continuous spectrum is real and independent of the perturbation and that the residual one is empty. We also show that a small perturbation can generate eigenvalues which converge to the threshold of continuous spectrum. We give the sufficient conditions for these eigenvalues to exist or to be absent and we construct the leading terms of their asymptotic expansions. We also describe the asymptotic behavior of the associated eigenfunctions. This is a joint work with D. Krejcirik.

Place: unusual: Horni Polubny
Date: unusual: Monday, January 22nd, morning session of the Goce's Winter school
Speaker: Petr Siegl (FJFI & DI)
Title and abstract: Quasi-Hermitian Hamiltonians
The definition of a Quasi-Hermitian operator together with basic spectral properties of the operator is presented. A general framework is supported by an example - H= -\frac{d^2}{dx^2} on (0,d) with PT-symmetric boundary conditions. An alternative construction without a metric operator \Theta is described and a connection between these two procedures is mentioned with help of the same example.

Date and time: 18. 1. 2007, 10.30 a.m.
Speaker: David Krejcirik (NPI & DI)
Title and abstract: Can one have a nodal line in 2D which does not touch the boundary?
An affirmative answer for Dirichlet Laplacian: the domain is simply connected but unbounded. Brief info on recent paper: P. Freitas and D. Krejcirik, Math. Res. Lett. 14 (2007) 107 - 111.

Date and time: 11. 1. 2007, 10.30 a.m.
Speaker: Tamas Fulop (NPI & DI)
Title and abstract: Singular potentials and ambiguity in the self-adjoint Hamiltonian
For a class of singular potentials, including the Coulomb potential and V(x) = g/x^2 with the coefficient g in a certain range, the corresponding Schrodinger operator is not automatically self-adjoint on its natural domain. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The talk discusses how the self-adjoint domains can be identified in terms of a boundary condition for the asymptotic behaviour of the wave functions around the singularity, and what physical differences emerge for different self-adjoint versions of the Hamiltonian.


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 secretaries of the microseminar were Vit Jakubsky and Miloslav Znojil

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