Weyl-Titchmarsh type formula for Hermite operator with small perturbation

TL;DR

This paper derives a Weyl-Titchmarsh type formula for the Hermite operator with small perturbations, linking polynomial asymptotics to spectral density and analyzing generalized eigenvectors.

Contribution

It introduces a Weyl-Titchmarsh type formula for perturbed Hermite operators and studies eigenvector bases with asymptotic behaviors.

Findings

Derived a spectral asymptotics formula for perturbed Hermite operators.

Established asymptotics of Plancherel-Rotach type for eigenvectors.

Connected polynomial asymptotics to spectral density in the perturbed setting.

Abstract

Small perturbations of the Jacobi matrix with weights \sqrt n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical Weyl-Titchmarsh formula for the Schroedinger operator on the half-line with summable potential. Additionally a base of generalized eigenvectors for "free" Hermite operator is studied and asymptotics of Plancherel-Rotach type are obtained.