# American Institute of Mathematical Sciences

June  2013, 6(3): 649-655. doi: 10.3934/dcdss.2013.6.649

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

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

Received  April 2010 Revised  January 2011 Published  December 2012

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 $L$.
Citation: Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649
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