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