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, 493–502 (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.
|
| Transparences: |
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.
|
| Transparences: |
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.
|
| Transparences: |
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:
http://de.arxiv.org/abs/quant-ph/0703118.
|
| 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.
|
Reminder:
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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