Eigenvalue dynamics of a PT-symmetric Sturm–Liouville operator and criteria for similarity to a self-adjoint or a normal operator

Abstract

The dynamics of the eigenvalues of a family of Sturm–Liouville operators with complex integrable PT-symmetric potential on a finite interval is studied. In the model case of the complex Airy operator, a criterion for the similarity of operators in the family to self-adjoint and normal operators is stated and the exceptional parameter values corresponding to multiple eigenvalues are analytically calculated.