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On self-adjointness of a Schrödinger operator
on differential forms

Author: Maxim Braverman
Journal: Proc. Amer. Math. Soc. 126 (1998), 617-623
MSC (1991): Primary 58G25; Secondary 35P05
MathSciNet review: 1443372
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Abstract: Let $M$ be a complete Riemannian manifold and let $\Omega ^{\bullet }(M)$ denote the space of differential forms on $M$. Let $d:\Omega ^{\bullet}(M)\to \Omega ^{\bullet+1}(M)$ be the exterior differential operator and let $\Delta =dd^{*}+d^{*}d$ be the Laplacian. We establish a sufficient condition for the Schrödinger operator $H=\Delta +V(x)$ (where the potential $V(x):\Omega ^{\bullet}(M)\to \Omega ^{\bullet}(M)$ is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by I. Oleinik about self-adjointness of a Schrödinger operator which acts on the space of scalar valued functions.

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Additional Information

Maxim Braverman
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics, The Hebrew University, Jerusalem 91904, Israel

Received by editor(s): August 19, 1996
Additional Notes: The research was supported by US - Israel Binational Science Foundation grant No. 9400299
Communicated by: Jeffrey B. Rauch
Article copyright: © Copyright 1998 American Mathematical Society