The Limit Spectral Graph in the Semi-Classical Approximation for the Sturm-Liouville Problem With a Complex Polynomial Potential

TL;DR

This paper investigates the asymptotic distribution of eigenvalues for Sturm-Liouville problems with complex polynomial potentials, revealing their concentration along a limit spectral graph and classifying its curves.

Contribution

It introduces a classification of the limit spectral graph and derives eigenvalue asymptotics along its curves for complex polynomial potentials.

Findings

Eigenvalues concentrate along the limit spectral graph at large parameters

The curves of the spectral graph are classified

Asymptotic formulas for eigenvalues along different curves are provided

Abstract

The limit distribution of the discrete spectrum of the Sturm-Liouville problem with complex-valued polynomial potential on an interval, on a half-axis, and on the entire axis is studied. It is shown that at large parameter values, the eigenvalues are concentrated along the so-called limit spectral graph; the curves forming this graph are classified. Asymptotics of eigenvalues along curves of various types in the graph are calculated.