Consider in the \Sc\ operator family depending on the real parameter , where is a complex-valued but symmetric periodic potential. An explicit condition on is obtained which ensures that the spectrum of 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.
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Sandro Graffi, Emanuela Caliceti, A criterion for the reality of the spectrum of <em>PT</em>-symmetric Schrödinger operators with complex-valued periodic potentials. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 2, pp. 163–173DOI 10.4171/RLM/515