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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

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 & 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.  Google Scholar

[2]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Pure and Applied Mathematics, 283, CRC Press, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.  Google Scholar

[7]

B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.  Google Scholar

[8]

J. Walter, Note on a paper by Stetkær-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.  Google Scholar

show all references

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.  Google Scholar

[2]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Pure and Applied Mathematics, 283, CRC Press, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness," Academic Press, New York, 1975.  Google Scholar

[7]

B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., 52 (1973), 44-48.  Google Scholar

[8]

J. Walter, Note on a paper by Stetkær-Hansen concerning essential self-adjointness of Schrödinger operators, Math. Scand., 25 (1969), 94-96.  Google Scholar

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