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arXiv · 2203.16889

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

Marco Bertola·Eduardo Chavez-Heredia·Tamara Grava
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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.

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Marco Bertola, Eduardo Chavez-Heredia, Tamara Grava. 2022-03-31. Exactly solvable anharmonic oscillator, degenerate orthogonal polynomials and Painleve' II. https://arxiv.org/abs/2203.16889

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