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.
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.
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.
After a brief introduction in the field, the one-dimensional Dirac Hamiltonian with square -well potential is discussed
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.
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.
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.