Opuscula Math. 29, no. 2 (2009), 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

Sergey Simonov

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.

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• Sergey Simonov
• Institute of Physics, St. Petersburg University, Department of Mathematical Physics, Ulianovskaia 1, 198904, St. Petergoff, St. Petersburg, Russia
• Received: 2008-06-13.
• Revised: 2008-12-30.
• Accepted: 2009-01-06.

Cite this article as:
Sergey Simonov, Weyl-Titchmarsh type formula for Hermite operator with small perturbation, Opuscula Math. 29, no. 2 (2009), 187-207, http://dx.doi.org/10.7494/OpMath.2009.29.2.187

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