Quantum star-graph analogues of PT-symmetric square wells. II: Spectra

TL;DR

This paper analyzes the spectra of non-Hermitian star-shaped quantum graphs with PT-symmetry, deriving explicit asymptotic formulas for their energy levels influenced by complex boundary conditions.

Contribution

It provides the first explicit asymptotic-expansion formulas for the spectra of PT-symmetric quantum star graphs with complex Robin boundary conditions.

Findings

Spectrum can be expressed in a closed asymptotic-expansion form.

Explicit formulas up to second order are derived.

Comments on the physical implications of the spectral formulas are included.

Abstract

For non-Hermitian equilateral q-pointed star-shaped quantum graphs of paper I [Can. J. Phys. 90, 1287 (2012), arXiv 1205.5211] we show that due to certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.