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
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.  Google Scholar

[2]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups,", Pure and Applied Mathematics, 283 (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.   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., 449 (1974), 182.   Google Scholar

[6]

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

[7]

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

[8]

J. Walter, Note on a paper by Stetkær-Hansen concerning essential self-adjointness of Schrödinger operators,, Math. Scand., 25 (1969), 94.   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.  doi: 10.2307/1999277.  Google Scholar

[2]

M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups,", Pure and Applied Mathematics, 283 (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.   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., 449 (1974), 182.   Google Scholar

[6]

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

[7]

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

[8]

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

[1]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[2]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[3]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[4]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[5]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[6]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[7]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[8]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[9]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[10]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[11]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[12]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[13]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[14]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[15]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[16]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[17]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[18]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[19]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[20]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]