Emanuela Caliceti, Sandro Graffi
A criterion for the reality of the spectrum of $\PT$ symmetric \Sc\ operators with complex-valued periodic potentials
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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.