Article Contents
Article Contents

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

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

 Citation:

•  [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.