Quantum graphs: self-adjoint, and yet exhibiting a nontrivial $\mathcal{PT}$-symmetry

TL;DR

This paper shows that quantum graphs can have a nontrivial $ ext{PT}$-symmetry when vertex matching conditions are circulant matrices, affecting their transport properties based on the coupling's structure.

Contribution

It introduces conditions for $ ext{PT}$-symmetry in quantum graphs using circulant matrices and explores how coupling components influence transport properties.

Findings

$ ext{PT}$-symmetry occurs with circulant vertex conditions

Transport properties depend on non-Robin coupling components

Nontrivial symmetry arises when matrices are not transposition-invariant

Abstract

We demonstrate that a quantum graph exhibits a $\mathcal{PT}$-symmetry provided the coefficients in the condition describing the wave function matching at the vertices are circulant matrices; this symmetry is nontrivial if they are not invariant with respect to transposition. We also illustrate how the transport properties of such graphs are significantly influenced by the presence or absence of the non-Robin component of the coupling.