Weyl-Titchmarsh type formula for Hermite operator with small perturbation

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 an analogue of the classical Weyl-Titchmarsh formula for the Schrödinger 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.