The form sum and the Friedrichs extension of Schrödinger-type operators on Riemannian manifolds

Author:
Ognjen Milatovic

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

Published electronically:
April 24, 2003

MathSciNet review:
2021257

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Abstract | References | Similar Articles | Additional Information

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.

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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

Article copyright:
© Copyright 2003
American Mathematical Society