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 groundstateenergy derivatives


In the MonteCarlo calculations of the groundstate 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 lowcost
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 PTsymmetric 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: 
EvaMaria Graefe (University of Kaiserslautern)

Title and abstract: 
The spectrum of an open twomode BoseHubbard system


In a twomode approximation a BoseEinstein condensate in a
doublewell trap can be described by a many particle Hamiltonian
of BoseHubbard type. We introduce effective complex energies to
the modes to describe a coupling to a continuum and obtain a
nonHermitian 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 nonHermiticity 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 BerryBarry 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 pseudoHermitian Hamiltonians


Brief review of results on PT symmetric and supersymmetric quantum
mechanical models will concentrate on a buildup of their
perturbative connection to positiondependentmass 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 modematching 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 doubleslit experiment


It is wellknown 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 noncommuting observables.
Recently, Ozawa (2003) reexamined the physical implication of the
uncertainty relation and formulated a new type of the uncertainty
relation. I examined a physical implication of the Young doubleslit
experiment to compare the roles of the KennardRobertson 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 eprint is available:
http://de.arxiv.org/abs/quantph/0703118.

Date and time: 
17. 5. 2007, 10.30 a. m. 
Speaker: 
Tamas Fulop (DI & NPI) 
Title and abstract: 
Physical role
attributed to selfadjointness in quantum mechanics


A
collection of pros and cons of keeping selfadjointness 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 PTsymmetric
quantum mechanics. We plan to show the aspects from which selfadjointness
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 PTSymmetric Oscillators. Journal of
Mathematical Physics, 42 (6): 25132530, 2001. ArXiv:
mathph/0007006
2. On the Reality of the Eigenvalues for a Class of PTSymmetric
Oscillators. Communications in Mathematical Physics, 229 (3):
543564, 2002. ArXiv: mathph/0201013
3. On halfline spectra for a class of nonselfadjoint Hill
operators. Mathematische Nachrichten, 261262: 171175, 2003. ArXiv: mathph/0308032
4. Eigenvalues of PTsymmetric oscillators with polynomial
potentials. Journal of Physics A: Mathematical and General, 38:
61476166, 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: hepph/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^2norm


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 PTsymmetric waveguide


We consider a planar waveguide of constant width with
PTsymmetric 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: 
QuasiHermitian Hamiltonians


The definition of a QuasiHermitian 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 PTsymmetric 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
selfadjoint 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 selfadjoint
on its natural domain. Such operators admit more than one selfadjoint
domain, and the spectrum and all physical consequences depend
seriously on the selfadjoint version chosen. The talk discusses how
the selfadjoint 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
selfadjoint versions of the Hamiltonian.

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 secretaries of the microseminar were 
Vit Jakubsky
and
Miloslav Znojil

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