A criterion for the reality of the spectrum of \(PT\)-symmetric Schrödinger operators with complex-valued periodic potentials (Q658523)

Summary: Consider in \(L^{2} (\mathbb{R})\) the Schrödinger 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.