arXiv subjects

Petr Siegl

Publications and source records attributed to Petr Siegl.

41 records · Page 3Linked to original sources

arXiv2010

Krein Spaces in de Sitter Quantum Theories

Experimental evidences and theoretical motivations lead to consider the curved space-time relativity based on the de Sitter group $SO_0(1,4)$ or $Sp(2,2)$ as an appealing substitute to the flat space-time Poincare relativity. Quantum elementary systems are then associated to unitary irreducible representations of that simple Lie group. At the lowest limit of the discrete series lies a remarkable family of scalar representations involving Krein structures and related undecomposable representation cohomology which deserves to be thoroughly studied in view of quantization of the corresponding carrier fields. The purpose of this note is to present the mathematical material needed to examine the problem and to indicate possible extensions of an exemplary case, namely the so-called de Sitterian massless minimally coupled field, i.e. a scalar field in de Sitter space-time which does not couple to the Ricci curvature.

hep-th↗

arXiv2010

PT-symmetric models in curved manifolds

We consider the Laplace-Beltrami operator in tubular neighbourhoods of curves on two-dimensional Riemannian manifolds, subject to non-Hermitian parity and time preserving boundary conditions. We are interested in the interplay between the geometry and spectrum. After introducing a suitable Hilbert space framework in the general situation, which enables us to realize the Laplace-Beltrami operator as an m-sectorial operator, we focus on solvable models defined on manifolds of constant curvature. In some situations, notably for non-Hermitian Robin-type boundary conditions, we are able to prove either the reality of the spectrum or the existence of complex conjugate pairs of eigenvalues, and establish similarity of the non-Hermitian m-sectorial operators to normal or self-adjoint operators. The study is illustrated by numerical computations.

math-ph↗

arXiv2009

Scattering in the PT-symmetric Coulomb potential

Scattering on the ${\cal PT}$-symmetric Coulomb potential is studied along a U-shaped trajectory circumventing the origin in the complex $x$ plane from below. This trajectory reflects ${\cal PT}$ symmetry, sets the appropriate boundary conditions for bound states and also allows the restoration of the correct sign of the energy eigenvalues. Scattering states are composed from the two linearly independent solutions valid for non-integer values of the 2L parameter, which would correspond to the angular momentum in the usual Hermitian setting. Transmission and reflection coefficients are written in closed analytic form and it is shown that similarly to other ${\cal PT}$-symmetric scattering systems the latter exhibit handedness effect. Bound-state energies are recovered from the poles of the transmission coefficients.

quant-ph↗

arXiv2009

Surprising spectra of PT-symmetric point interactions

Spectra of the second derivative operators corresponding to the special PT-symmetric point interactions are studied. The results are partly the completion of those obtained in [1]. The particular PT-symmetric point interactions causing unusual spectral effects are investigated for the systems defined on a finite interval as well. The spectrum of this type of interactions is very far from the self-adjoint case despite of PT-symmetry, P-pseudo-Hermiticity and T-self-adjointness.

math-ph↗

arXiv2009

Asymptotically vanishing PT-symmetric potentials and negative-mass Schroedinger equations

In paper I [M. Znojil and G. Lévai, Phys. Lett. A 271 (2000) 327] we introduced the Coulomb - Kratzer bound-state problem in its cryptohermitian, PT-symmetric version. An instability of the original model is revealed and its necessary stabilization is achieved, for almost all couplings, by an unusual, negative choice of the bare mass in Schroediner equation.

quant-ph↗