Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-98-04284-1
MathSciNet review: 1443372
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [BS] F. A. Berezin, M. A. Shubin, The Schrödinger equation, Kluwer, Dordrecht, 1991. MR 93i:81001
  • [Ch] P. R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. of Functional Analysis 12 (1973), 401-414. MR 51:6119
  • [Ga1] M. P. Gaffney, The harmonic operators for exterior differential forms, Proc. Nat. Acad. Sci. USA 37 (1951), 48-50. MR 13:987b
  • [Ga2] M. P. Gaffney, A special Stokes's theorem for complete Riemannian manifolds, Ann. of Math. 60 (1954), 140-145. MR 15:986d
  • [Le] B. M. Levitan, On a theorem of Titchmarsh and Sears, Usp. Math. Nauk 16 (1961), 175-178. MR 24:A2133
  • [O1] I. M. Oleinik, On the essential self-adjointness of the Schrödinger operator on a complete Riemannian manifold, Mathematical Notes 54 (1993), 934- 939. MR 94m:58226
  • [O2] I. M. Oleinik, On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete Riemannian manifolds, Mathematical Notes 55 (1994), 380-386. MR 95h:35051
  • [RB] F. S. Rofe-Beketov, Self-adjointness conditions for the Schrödinger operator, Mat. Zametki 8 (1970), 741-751. MR 43:743
  • [RS] M. Reed, B. Simon, Methods of modern mathematical physics, Vol. I, II, Academic Press, London, 1972, 1975. MR 58:12429a; MR 58:12429b
  • [Se] D. B. Sears, Note on the uniqueness of Green's functions associated with certain differential equations, Canadian J. Math. 2 (1950), 314-325. MR 12:102h
  • [Sh] M. A. Shubin, Spectral theory of elliptic operators on non-compact manifolds, Astérisque 207 (1992), 37-108. MR 94h:58175
  • [Wa] F. W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, Springer-Verlag, New-York, Berlin, Heidelberg, Tokyo, 1983. MR 84k:58001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58G25, 35P05

Retrieve articles in all journals with MSC (1991): 58G25, 35P05


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
Email: maxim@math.ohio-state.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04284-1
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

American Mathematical Society