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

E. Caliceti; S. Graffi

paper

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

arXiv:0804.4578

Abstract

Consider in $L^{2} (R)$ the \Sc operator family $H (g) := - d_{x}^{2} + V_{g} (x)$ depending on the real parameter $g$ , where $V_{g} (x)$ is a complex-valued but $P T$ 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.