Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 106, 13 pages      arXiv:1803.06819      https://doi.org/10.3842/SIGMA.2018.106
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators

Victor Yu. Novokshenov
Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Str., 450008, Ufa, Russia

Received March 20, 2018, in final form September 20, 2018; Published online September 30, 2018

Abstract
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.

Key words: generalized Hermite polynomials; monodromy-free Schrödinger operator; Painlevé IV equation; meromorphic solutions; distribution of zeros; 1D SUSY quantum mechanics.

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