PT-symmetric models in curved manifolds

TL;DR

This paper investigates PT-symmetric Laplace-Beltrami operators on curved manifolds, analyzing their spectral properties and demonstrating conditions for real spectra or complex conjugate eigenvalues, with numerical support.

Contribution

It introduces a Hilbert space framework for PT-symmetric operators on curved manifolds and analyzes spectral properties in constant curvature cases, including non-Hermitian boundary conditions.

Findings

Spectrum can be real or form complex conjugate pairs.

Operators are similar to normal or self-adjoint operators under certain conditions.

Numerical computations support theoretical results.

Abstract

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.