Quasi-exactly solvable quartic: elementary integrals and asymptotics

TL;DR

This paper investigates elementary eigenfunctions of certain differential operators with polynomial coefficients, deriving identities to describe the spectral locus and analyzing its asymptotic behavior for a specific class of operators.

Contribution

The paper introduces a new identity for operators with odd cubic polynomial h, advancing understanding of the spectral locus and its asymptotics in quasi-exactly solvable quartic problems.

Findings

Derived an identity for spectral analysis when h is an odd cubic polynomial.

Described the spectral locus for the class of operators studied.

Established asymptotic properties of the QES spectral locus.

Abstract

We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we found an interesting identity which is used to describe the spectral locus. We also establish some asymptotic properties of the QES spectral locus.