Opuscula Math. 29, no. 2 (), 187-207
http://dx.doi.org/10.7494/OpMath.2009.29.2.187
Opuscula Mathematica

# 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.
Keywords: Jacobi matrices, absolutely continuous spectrum, subordinacy theory, Weyl-Titchmarsh theory.
Mathematics Subject Classification: 47A10, 47B36.