# Quantum Graphs: $ \mathcal{PT}$-symmetry and reflection symmetry of the spectrum

## TL;DR

This paper investigates the relationship between spectral symmetry and geometric automorphisms in non-self-adjoint quantum graphs, showing that spectral reflection symmetry implies $\, ext{ extit{PT}}$-symmetry for equilateral star-graphs.

## Contribution

It establishes a link between spectral symmetry and $\, ext{ extit{PT}}$-symmetry in quantum graphs, specifically proving the equivalence for equilateral star-graphs.

## Key findings

- Spectral reflection symmetry implies $\, ext{ extit{PT}}$-symmetry for equilateral star-graphs.
- The Laplace operator with Robin conditions has symmetric spectrum iff it is $\, ext{ extit{PT}}$-symmetric.
- The results connect geometric automorphisms with spectral properties in non-self-adjoint quantum graphs.

## 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) $ \mathcal P $. If the differential operator is $ \mathcal P \mathcal T$-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 $ \mathcal P \mathcal T$-symmetric. We give partial answer to this question by considering equilateral star-graphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is $ \mathcal P \mathcal T$-symmetric with $ \mathcal P $ being an automorphism of the metric graph.