Group Analysis of Differential Equations and Integrable Systems − 2018
(Institute of Mathematics of Russian Academy of Sciences, Ufa Federal Research Center, Ufa, Russia)
Generalized Hermite polynomials and monodromy-free potentials
We consider a class of monodromy-free Schrödinger operators with rational 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 origin. We derive explicit condition of non-singularity of 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 dressing of 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 Painlevé IV equation. We discuss the structure of discrete spectrum of Schrödinger operators and its link to monodromy-free condition.