# 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 manifoldsHTML articles powered by AMS MathViewer

by Ognjen Milatovic
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.
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