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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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.
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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:,
  • 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:
  • MathSciNet review: 2021257