Eigenvalue Dynamics of a PT-symmetric Sturm-Liouville Operator. Criteria of the Similarity to a Self-adjoint or Normal Operator

The goal of the paper is to investigate the dynamics of the eigenvalues of the Sturm-Liouville operator with summable PT-symmetric potential on the finite interval. It turns out that the case of a complex Airy operator presents an exactly solvable model which allows us to trace the dynamics of the movement of the eigenvalues in all details and to find explicitly the critical parameter values, in particular, to specify precisely the number $\varepsilon_1$ such that for $0<\varepsilon<\varepsilon_1$ the operator has a real spectrum and is similar to a self-adjoint operator.