## Speakers

Denis Borisov (Bashkir State Pedagogical University, Ufa)

Vit Jakubsky, Hynek Bila and Milos Tater (Institute of Nuclear Physics, Rez)

Uwe Guenther (Research Center Dresden-Rossendorf)

Geza Levai (Institute of Nuclear Research, Debrecen)

Denis Kochan (Comenius University, Bratislava)

Artur Sergyeyev (Silesian University, Opava)

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## The lectures' final timetable, in pdf, is here Topics:

Denis Borisov: A PT-symmetric waveguide

Vit Jakubsky: Physical systems described by pseudo-Hermitian operators

Uwe Guenther: original title:
The spherically symmetric \alpha2-dynamo, resonant unfolding of diabolical points and third-order exceptional points in Krein space related setups.
final title:
Results on Krein space related $\alpha^2-$dynamos

Hynek Bila: original title:
Non-Hermitian operators: models and methods
final title:
How does the spectrum complexify?

Geza Levai: PT symmetry and non-central potentials in 2 and 3 dimensions

Milos Tater: Root Asymptotics of Spectral Polynomials

Denis Kochan: New method of quantization of dissipative systems

Artur Sergyeyev: Flat coordinates and hidden symmetry for Benenti systems

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## Abstracts

Denis Borisov:
We consider a planar waveguide of constant with PT-symmetric 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 pseudo-hermitian Hamiltonian. We show that the explicit knowledge of the non-trivial 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^2-dynamo with idealized boundary conditions. The corresponding operator is self-adjoint in a Krein space and therefore it shares many features with Hamiltonians of PT-symmetric Quantum Mechanics. The spectrum of a dynamo with constant \alpha-profile 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 second-order exceptional points in Krein space related models and the emergence of third-order Jordan structures. We demonstrate the basic mechanism on a most simple PT-symmetric 4x4 matrix model and use the obtained results to identify similar structures in the spectral decomposition of \alpha^2-dynamo operators. A joint work with Oleg Kirillov and Frank Stefani.

previous version:

In the first part of the talk we consider the spectral behavior of the spherically symmetric \alpha2-dynamo with idealized boundary conditions. The corresponding operator is self-adjoint in a Krein space and therefore it shares many features with Hamiltonians of PT-symmetric Quantum Mechanics. The spectrum of a dynamo with constant \alpha-profile 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 second-order exceptional points in Krein space related models and the emergence of third-order Jordan structures. We demonstrate the basic mechanism on a most simple PT-symmetric 4x4 matrix model and use the obtained results to identify similar structures in the spectral decomposition of \alpha2-dynamo operators. A joint work with Oleg Kirillov and Frank Stefani.

Hynek Bila:
After a brief introduction in the field, the one-dimensional 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 non-degenerate 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 PT-symmetric 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 one-dimensional PT-symmetric potentials (e.g. indefinite pseudo-norm, quasi-parity) and multi-dimensional 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 potential-generated will be presented. Time evolution in that case is governed by certain canonical two-form $\Omega$ (an analog of $dp/\dq-dH/\dt$), 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 two-form $\Omega$ will be proposed. The Feynman's path integral over histories of the system will be rearranged to a umbilical world-sheet'' functional integral. In the special case of potential-generated forces, world-sheet'' 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.

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## Registration

optional, not necessary

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## next meeting

April 3rd, 2007, Villa Lanna, 9.30 a.m.,
currently under preparation

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## contacts

e-mail:
znojil@ujf.cas.cz

letter:
Miloslav Znojil
Nuclear Physics Institute,250 68 Rez ,Czech Republic

FAX:
+420 2 20940165

phone:
+420 2 6617 3286 or +420 724 747 898

February 21th, 2007, final version, by Miloslav Znojil                            return upwards                            return to the webpage of microseminars