Spectral Deformation in a Problem of Singular Perturbation Theory

Abstract—Quasi-classical asymptotic behavior of the spectrum of a non-self-adjoint Sturm–Liouville problem is studied in the case of a one-parameter family of potentials being third-degree polynomials. For this problem, the phase-integral method is used to derive quantization conditions characterizing the asymptotic distribution of the eigenvalues and their concentration near edges of the limit spectral complex. Topologically different types of limit configurations are described, and critical values of the deformation parameter corresponding to type changes are specified.