Abstracts
Denis Borisov:
We consider a planar waveguide of constant with PTsymmetric Robin
boundary conditions. We 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 is independent
of such perturbation and the residual one is empty. We also
consider a small perturbation and show that it can originates the
eigenvalues those converge to the threshold of the continuous
spectrum. We give the sufficient conditions these eigenvalues to
exist or to be absent and 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.
Vit Jakubsky:
We discuss the ways how to make physical predictions on
a system described by a pseudohermitian Hamiltonian. We show that
the explicit knowledge of the nontrivial scalar product is not
essential in some cases.
Uwe Guenther: In the first part of the talk we consider the spectral behavior of the spherically
symmetric \alpha^2dynamo with idealized boundary conditions. The corresponding
operator is selfadjoint in a Krein space and
therefore it shares many features with Hamiltonians of
PTsymmetric Quantum Mechanics. The spectrum of a dynamo with constant
\alphaprofile contains a countably infinite number of diabolical points
which under inhomogeneous perturbations unfold in a very specific and resonant way.
We describe this mechanism in detail and discuss its physical implications.
In the second part of the talk we discuss coalescing secondorder exceptional points
in Krein space related models and the emergence of thirdorder Jordan structures. We
demonstrate the basic mechanism on a
most simple PTsymmetric 4x4 matrix model and use the obtained results to identify
similar structures in the spectral decomposition of \alpha^2dynamo operators.
A joint work with Oleg Kirillov and Frank Stefani.
Hynek Bila:
After a brief introduction in the field,
the onedimensional Dirac Hamiltonian with square well potential is discussed
Milos Tater:
In the first part of the talk we review results concerning the
algebraic part of multiparameter spectral problem for higher Lame equation
in the nondegenerate case. We focus on root localization of Van Vleck and
Stieltjes polynomials. In the second part, we open some problems in
degenerate case, which bear upon QES Schrodinger operators.
Geza Levai: We investigate the conditions under
which PTsymmetric
potentials can be solved exactly in 2 and 3 dimensions by the
separation of the radial and angular coordinates. The possible
occurrence of specific properties characterizing onedimensional
PTsymmetric potentials (e.g. indefinite pseudonorm, quasiparity)
and multidimensional real central potentials (e.g. degeneracy
patterns, algebraic structures) are also discussed in this general
framework.
Denis Kochan:
Geometrical formulation of classical mechanics with forces that are not
necessarily
potentialgenerated will be presented. Time evolution in that case is
governed by
certain canonical twoform $\Omega$ (an analog of $dp/\dqdH/\dt$), which is
constructed
purely from forces and the metric tensor entering the kinetic energy of the
system.
Attempt to ``dissipative quantization'' in terms of the twoform
$\Omega$ will be proposed.
The Feynman's path integral over histories of the system will be rearranged
to a ``umbilical worldsheet'' functional integral. In the special case of
potentialgenerated forces, ``worldsheet'' approach precisely reduces to
the standard quantum mechanics. However, a transition
probability amplitude expressed in terms of ``string functional
integral'' can be applicable (at
least academically) when a general dissipative environment is discussed.
