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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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The form sum and the Friedrichs extension of Schrödinger-type operators on Riemannian manifolds
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by Ognjen Milatovic PDF
Proc. Amer. Math. Soc. 132 (2004), 147-156 Request permission

Abstract:

We consider $H_V=\Delta _M+V$, where $(M,g)$ is a Riemannian manifold (not necessarily complete), and $\Delta _M$ is the scalar Laplacian on $M$. We assume that $V=V_0+V_1$, where $V_0\in L_{\operatorname {loc}}^2(M)$ and $-C\leq V_1\in L_{\operatorname {loc}}^1(M)$ ($C$ is a constant) are real-valued, and $\Delta _M+V_0$ is semibounded below on $C_{c}^{\infty }(M)$. Let $T_0$ be the Friedrichs extension of $(\Delta _M+V_0)|_{C_{c}^{\infty }(M)}$. We prove that the form sum $T_0\tilde {+} V_1$ coincides with the self-adjoint operator $T_F$ associated to the closure of the restriction to $C_{c}^{\infty }(M)\times C_{c}^{\infty }(M)$ of the sum of two closed quadratic forms of $T_0$ and $V_1$. This is an extension of a result of Cycon. The proof adopts the scheme of Cycon, but requires the use of a more general version of Kato’s inequality for operators on Riemannian manifolds.
References
  • M. Braverman, O. Milatovic, M. A. Shubin, Essential self-adjointness of Schrödinger type operators on manifolds, Russian Math. Surveys, 57 (4) (2002), 641–692.
  • Hans L. Cycon, On the form sum and the Friedrichs extension of Schrödinger operators with singular potentials, J. Operator Theory 6 (1981), no. 1, 75–86. MR 637002
  • William G. Faris, Self-adjoint operators, Lecture Notes in Mathematics, Vol. 433, Springer-Verlag, Berlin-New York, 1975. MR 0467348
  • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
  • Heinz-Willi Goelden, On non-degeneracy of the ground state of Schrödinger operators, Math. Z. 155 (1977), no. 3, 239–247. MR 609535, DOI 10.1007/BF02028443
  • H. Hess, R. Schrader, and D. A. Uhlenbrock, Domination of semigroups and generalization of Kato’s inequality, Duke Math. J. 44 (1977), no. 4, 893–904. MR 458243
  • H. Hess, R. Schrader, and D. A. Uhlenbrock, Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds, J. Differential Geometry 15 (1980), no. 1, 27–37 (1981). MR 602436
  • Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
  • Tosio Kato, A second look at the essential selfadjointness of the Schrödinger operators, Physical reality and mathematical description, Reidel, Dordrecht, 1974, pp. 193–201. MR 0477431
  • Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
  • Barry Simon, An abstract Kato’s inequality for generators of positivity preserving semigroups, Indiana Univ. Math. J. 26 (1977), no. 6, 1067–1073. MR 461209, DOI 10.1512/iumj.1977.26.26086
  • Barry Simon, Kato’s inequality and the comparison of semigroups, J. Functional Analysis 32 (1979), no. 1, 97–101. MR 533221, DOI 10.1016/0022-1236(79)90079-X
  • Barry Simon, Maximal and minimal Schrödinger forms, J. Operator Theory 1 (1979), no. 1, 37–47. MR 526289
  • Michael E. Taylor, Partial differential equations. II, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996. Qualitative studies of linear equations. MR 1395149, DOI 10.1007/978-1-4757-4187-2
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Additional Information
  • Ognjen Milatovic
  • Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
  • Address at time of publication: Department of Mathematics, Fitchburg State College, Fitchburg, Massachusetts 01420
  • MR Author ID: 705360
  • Email: ogmilato@lynx.neu.edu, omilatovic@fsc.edu
  • Received by editor(s): August 20, 2002
  • Published electronically: April 24, 2003
  • Communicated by: David S. Tartakoff
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 147-156
  • MSC (2000): Primary 35P05, 58G25; Secondary 47B25, 81Q10
  • DOI: https://doi.org/10.1090/S0002-9939-03-07075-8
  • MathSciNet review: 2021257