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

Abstract

Consider in $L^2(\mathbb{R})$ the Schrödinger 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 $PT$ 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.