A criterion for the reality of the spectrum of PT symmetric Schroedinger operators with complex-valued periodic potentials

TL;DR

This paper establishes explicit criteria for PT symmetric Schrödinger operators with complex periodic potentials to have purely real spectra or to contain complex arcs, advancing understanding of spectral properties in non-Hermitian quantum mechanics.

Contribution

It provides new explicit conditions on the potential that determine when the spectrum is real or contains complex arcs, enhancing spectral analysis of PT symmetric operators.

Findings

Spectrum is purely real under certain conditions.

Spectrum can contain complex analytic arcs.

Explicit criteria for spectral reality and complexity.

Abstract

Consider in $L^2(\R)$ the \Sc operator family $H(g):=-d^2_x+V_g(x)$ depending on the real parameter $g$, where $V_g(x)$ is a complex-valued but $PT$ symmetric periodic potential. An explicit condition on $V$ is obtained which ensures that the spectrum of $H(g)$ is purely real and band shaped; furthermore, a further condition is obtained which ensures that the spectrum contains at least a pair of complex analytic arcs.