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Essential self-adjointness of Schrödinger-type operators on manifolds
M. Braverman,
O. Milatovica,
M. A. Shubina a Northeastern University
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.
UDC:
517.956.2+
517.984
MSC: Primary
47B25,
35J10; Secondary
53C20,
34L40,
81Q10,
58B20 Received: 31.03.2002
DOI:
10.4213/rm532
Fulltext:
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