Weyl-Titchmarsh type formula for Hermite operator with small perturbation

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.

Abstract licence: CC BY

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