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Essential self-adjointness of Schrödinger-type operators on manifolds. (English. Russian original) Zbl 1052.58027

Russ. Math. Surv. 57, No. 4, 641-692 (2002); translation from Usp. Mat. Nauk 57, No. 4, 3-58 (2002).
The main goal of the very interesting paper under review is to obtain sufficient conditions ensuring 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 Hermitean vector bundle \(E\) over a manifold \(M\) with positive smooth measure \(d\mu\) and \(V\) is a Hermitean bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on \(M\) associated with \(H_V.\) The manifold \(M\) is not assumed a priori to be endowed with a complete Riemannian metric but it is constructed from the operator. This enables one to treat, for instance, operators acting on bounded domains in \({\mathbb R}^n\) with the Lebesgue measure. Another kind of results is established by considering singular potentials \(V\) through application of a refined Kato-type inequality on vector bundles. 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. In the special case when the principal symbol of \(D^*D\) is scalar, more precise results are derived for operators with singular potentials.
The approach developed by the authors unifies and extends almost all earlier results on essential self-adjointness of Schrödinger-type operators on vector bundles over manifolds.
The exposition is excellent and self-contained and the reader will surely appreciate a lot of history on the subject and the open problems proposed.

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35J10 Schrödinger operator, Schrödinger equation
47B25 Linear symmetric and selfadjoint operators (unbounded)
53C20 Global Riemannian geometry, including pinching
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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