Bifurcation of spectra of periodic self-adjoint operators on axis with small 𝒫𝒯-symmetric potential

Abstract We consider an one-dimensional Schrödinger operator with an even periodic potential perturbed by a small <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒫</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">𝒯</m:mi> </m:mrow> </m:math> {\mathcal{PT}} -symmetric potential. The unperturbed spectrum can have points associated with two periodic or antiperiodic eigenfunctions. We study how such points bifurcate under the perturbation. We obtain a three-terms asymptotic expansion for the corresponding perturbed band functions with rigorous estimates for the error terms. These asymptotics allow us to establish sufficient conditions for the emergence and absence of a non-real spectrum. The emerging non-real spectrum is a complex curve, the shape of which is described by our asymptotic expansions. We also describe how the Dirichlet eigenavules, which satisfy the Dubrovin equations for finite-gap potentials, depend on the perturbation.