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
References:
[1]

P. Baras and J. A. Goldstein, The heat equation with singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. doi: 10.2307/1999277.

[2]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups,", Pure and Applied Mathematics, 283 (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.

[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., 449 (1974), 182.

[6]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,", Academic Press, (1975).

[7]

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

[8]

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

show all references

References:
[1]

P. Baras and J. A. Goldstein, The heat equation with singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. doi: 10.2307/1999277.

[2]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups,", Pure and Applied Mathematics, 283 (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.

[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., 449 (1974), 182.

[6]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,", Academic Press, (1975).

[7]

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

[8]

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

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