{\mathcal {PT}}-symmetric models in curved manifolds

David Krejčiřík, Petr Siegl
2010 Journal of Physics A: Mathematical and Theoretical  
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.
more » ... onstant 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.
doi:10.1088/1751-8113/43/48/485204 fatcat:euhlog5e6bgzndgwl4lzuruiqq