Puglia Summer Trimester 2023 - Singularities, Asymptotics and Limiting Models - Seminar

Speaker: Marco Bertola (Concordia University - Montreal)

Title: Anharmonic oscillators, Fekete points and a conjecture of Shapiro and Tater

Abstract: In this talk I will discuss the recent solution of a conjecture of Shapiro and Tater about the location of the zeroes of discriminants for the secular equation of a particular anharmonic oscillator. It was a numerical observation that these zeroes form almost the same pattern as the poles of the rational solutions of the second Painlevé equation. In our recent work (in progress) with T. Grava and E. C. Heredia we show how to quantify the similarity between the two patterns. In the process we also connect the eigenfunctions of the "Exactly Solvable" spectrum of a quartic anharmonic oscillator with "degenerate" orthogonal polynomials. This connection shows to be surprisingly fruitful and extends to more general situations, allowing to generalize the celebrated theorem of Stieltjes on the electrostatic interpretation of the zeroes of the classical orthogonal polynomials.

For other seminars in the PST2023 seminar series, click https://sites.google.com/view/pst2023/visiting-scientists