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Russian Mathematical Surveys, 2002, Volume 57, Issue 4, Pages 641–692
DOI: https://doi.org/10.1070/RM2002v057n04ABEH000532
(Mi rm532)
 

This article is cited in 95 scientific papers (total in 96 papers)

Essential self-adjointness of Schrödinger-type operators on manifolds

M. Braverman, O. Milatovica, M. A. Shubina

a Northeastern University
References:
Abstract: Several conditions are obtained ensuring the essential self-adjointness of a Schrödinger-type operator $H_V=D^*D+V$, where $D$ is a first-order elliptic differential operator acting on the space of sections of a Hermitian vector bundle $E$ over a manifold $M$ with positive smooth measure $d\mu$ and $V$ is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on $M$ naturally associated with $H_V$. The results generalize theorems of Titchmarsh, Sears, Rofe-Beketov, Oleinik, Shubin, and Lesch. It is not assumed a priori that $M$ is endowed with a complete Riemannian metric. This enables one to treat, for instance, operators acting on bounded domains in $\mathbb R^n$ with Lebesgue measure. Singular potentials $V$ are also admitted. In particular, a new self-adjointness condition is obtained for a Schrödinger operator on $\mathbb R^n$ whose potential has a Coulomb-type singularity and can tend to $-\infty$ at infinity. For the special case in which the principal symbol of $D^*D$ is scalar, more precise results are established for operators with singular potentials. The proofs of these facts are based on a refined Kato-type inequality modifying and improving a result of Hess, Schrader, and Uhlenbrock.
Received: 31.03.2002
Bibliographic databases:
Document Type: Article
UDC: 517.956.2+517.984
MSC: Primary 47B25, 35J10; Secondary 53C20, 34L40, 81Q10, 58B20
Language: English
Original paper language: Russian
Citation: M. Braverman, O. Milatovic, M. A. Shubin, “Essential self-adjointness of Schrödinger-type operators on manifolds”, Russian Math. Surveys, 57:4 (2002), 641–692
Citation in format AMSBIB
\Bibitem{BraMilShu02}
\by M.~Braverman, O.~Milatovic, M.~A.~Shubin
\paper Essential self-adjointness of Schr\"odinger-type operators on manifolds
\jour Russian Math. Surveys
\yr 2002
\vol 57
\issue 4
\pages 641--692
\mathnet{http://mi.mathnet.ru//eng/rm532}
\crossref{https://doi.org/10.1070/RM2002v057n04ABEH000532}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1942115}
\zmath{https://zbmath.org/?q=an:1052.58027}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2002RuMaS..57..641B}
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  • This publication is cited in the following 96 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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