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

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Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator

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

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), Italy, Italy

Received April 2010 Revised January 2011 Published December 2012

Abstract
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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 $L$.

Keywords: positivity preserving $C_0$-semigroup, Ornstein-Uhlenbeck operator, Inverse square potential, Hardy's inequality, essential self-adjointness, Kolmogorov operator..

Mathematics Subject Classification: Primary: 47B25, 47D06, 35K65, 35R05; Secondary: 35K15, 35B25, 34G1.

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

References:

[1]	P. Baras and J. A. Goldstein, The heat equation with singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. doi: 10.2307/1999277.
[2]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Pure and Applied Mathematics, 283, CRC Press, 2006.
[3]	G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Applicable Analysis. doi: 10.1080/00036811.2011.587809.
[4]	T. Ikebe and T. Kato, Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Rational Mech. Anal., 9 (1962), 77-92.
[5]	H. Kalf, U. W. Schmincke, J. Walter and R. Wüst, "On the Spectral Theory of Schrödinger and Dirac Operators with Strongly Singular Potentials," Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., 449, 182-226. Springer, Berlin, 1975.
[6]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.
[7]	B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.
[8]	J. Walter, Note on a paper by Stetkær-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.

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References:

[1]	P. Baras and J. A. Goldstein, The heat equation with singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. doi: 10.2307/1999277.
[2]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Pure and Applied Mathematics, 283, CRC Press, 2006.
[3]	G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Applicable Analysis. doi: 10.1080/00036811.2011.587809.
[4]	T. Ikebe and T. Kato, Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Rational Mech. Anal., 9 (1962), 77-92.
[5]	H. Kalf, U. W. Schmincke, J. Walter and R. Wüst, "On the Spectral Theory of Schrödinger and Dirac Operators with Strongly Singular Potentials," Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., 449, 182-226. Springer, Berlin, 1975.
[6]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.
[7]	B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.
[8]	J. Walter, Note on a paper by Stetkær-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.

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