On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials

Tiziana Durante; Abdelaziz Rhandi

doi:10.3934/dcdss.2013.6.649

Article Contents

2013, Volume 6, Issue 3: 649-655. Doi: 10.3934/dcdss.2013.6.649

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On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials

Tiziana Durante^1, and
Abdelaziz Rhandi^1,

1.
Dipartimento di Ingegneria dell'Informazione, Ingegneria Elletrica e Matematica Applicata, Università degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (Sa)

Received: April 2010

Revised: January 2011

Published: June 2013

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Abstract

In this note we give sufficient conditions for the essential self-adjointness of some Kolmogorov operators perturbed by singular potentials. As an application we show that the space of test functions $C_c^∞(R^N \backslash \{0\})$ is a core for the operator $Au= Δu-Bx∇u+\frac{c}{|x|^2} u=:Lu+\frac{c}{|x|^2} u, u ∈ C_c^∞(R^N \backslash \{0\}),$ in $L^2(R^N,\mu)$ provided that $c\le \frac{(N-2)^2}{4}-1$. Here $B$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$.

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Mathematics Subject Classification: Primary: 47B25, 47D06, 35K65, 35R05; Secondary: 35K15, 35B25, 34G10.

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