Quantum graphs: PT -symmetry and reflection symmetry of the spectrum

Abstract

Not necessarily self-adjoint quantum graphs-differential operators on metric graphs-are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) P. If the differential operator is PT -symmetric, then its spectrum has reflection symmetry with respect to the real line. Our goal is to understand whether the opposite statement holds, namely, whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is PT - symmetric. We give partial answer to this question by considering equilateral stargraphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is PT -symmetric with P being an automorphism of the metric graph.