Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Author: Novokshenov, V.Yu.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209851
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators / V.Yu. Novokshenov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ.

The paper gives a review of recent progress in the classification of monodromy-free Schrödinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials, defined as Wronskians of classic Hermite polynomials, appear in a number of mathematical physics problems as well as in the theory of random matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those potentials demonstrate localized oscillatory behavior near the origin. We derive an explicit condition of non-singularity of the corresponding potentials and estimate a localization range with respect to indices of polynomials and distribution of their zeros in the complex plane. It turns out that 1D SUSY quantum non-singular potentials come as a dressing of the harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this end, all generalized Hermite polynomials are produced by appropriate periodic closure of this algebra, which leads to rational solutions of the Painlevé IV equation. We discuss the structure of the discrete spectrum of Schrödinger operators and its link to the monodromy-free condition.