PT-symmetric models in curved manifolds

0351846 - ÚJF 2011 RIV GB eng J - Journal Article
Krejčiřík, David - Siegl, Petr
PT-symmetric models in curved manifolds.
Journal of Physics A-Mathematical and Theoretical. Roč. 43, č. 48 (2010), 485204/1-485204/30. ISSN 1751-8113. E-ISSN 1751-8121
R&D Projects: GA MŠMT LC06002
Institutional research plan: CEZ:AV0Z10480505
Keywords : NON-HERMITIAN HAMILTONIANS * SCHRODINGER-TYPE OPERATORS * PSEUDO-HERMITICITY
Subject RIV: BA - General Mathematics
Impact factor: 1.641, year: 2010

We consider the Laplace-Beltrami operator in tubular neighborhoods 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-Hermitianm-sectorial operators to normal or self-adjoint operators. The study is illustrated by numerical computations.
Permanent Link: http://hdl.handle.net/11104/0191502