# Exactly Solvable Anharmonic Oscillator, Degenerate Orthogonal Polynomials and Painlevé II

## TL;DR

This paper explores a mathematical connection between an anharmonic oscillator and special polynomials using WKB analysis.

## Contribution

The paper reveals a new link between anharmonic oscillators and degenerate orthogonal polynomials through WKB analysis.

## Key findings

- The set of t-values with repeated eigenvalues matches the zeroes of Vorob’ev–Yablonskii polynomials.
- A deep connection is found between anharmonic oscillators and degenerate orthogonal monic polynomials.

## Abstract

Using WKB analysis, the paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the set the values of \documentclass[12pt]{minimal}
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\begin{document}$$t\in \mathbb {C}$$\end{document}t∈C for which the spectrum of the quartic anharmonic oscillator in the complex plane \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{\textrm{d}^{2} y}{\textrm{d} x^{2}} - \left( x^4 + tx^2 + 2Jx \right) y = \Lambda y, \end{aligned}$$\end{document}d2ydx2-x4+tx2+2Jxy=Λy,with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob’ev–Yablonskii polynomials, i.e. the poles of rational solutions of the second Painlevé equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal (monic) polynomials.