Exactly solvable anharmonic oscillator, degenerate orthogonal polynomials and Painleve' II

TL;DR

This paper explores a conjecture linking the spectral properties of a complex quartic anharmonic oscillator to the zeros of special polynomials related to the Painlevé II equation, revealing deep mathematical connections.

Contribution

It establishes a surprising connection between the spectrum of a complex anharmonic oscillator and the zeros of Vorob'ev-Yablonskii polynomials, related to Painlevé II solutions.

Findings

Confirmed the conjecture relating oscillator spectra and polynomial zeros

Identified a link between anharmonic oscillators and degenerate orthogonal polynomials

Uncovered a deep mathematical relationship between spectral theory and Painlevé equations

Abstract

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 values of $t\in \mathbb{C}$ for which the spectrum of the quartic anharmonic oscillator in the complex plane $$\frac{{\rm d}^2 y}{{\rm d}x^2} - ( x^4 + tx^2 + 2Jx )y = \Lambda y, $$ 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 Painlev\'e equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal polynomials.