Library of Congress Cataloging-in•Publication Data Shubin, M. A. (Mikhai1 Aleksandrovich), 1944-[psevdodifferentsial'nye operatory i spektral'naia teoriia. Englishj Pseudodifferential operators and spectral theory I M.A. Shubin ; translated from the Russian by Stig 1. Andersson.-2nd ed. p.em.

The series is divergent; therefore we may be able to do something with it.

Library of Congress Cataloging-in•Publication Data Shubin, M. A. (Mikhai1 Aleksandrovich), 1944-[psevdodifferentsial'nye operatory i spektral'naia teoriia. Englishj Pseudodifferential operators and spectral theory I M.A. Shubin ; translated from the Russian by Stig 1. Andersson.-2nd ed. p.em.

Abstract. We obtain several essential self-adjointness conditions for the Schrödinger type operator HV = 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 dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV . These results generalize the theorems of E. C. Titchmarsh, D. B. Sears, F. S. Rofe-Beketov, I. M. Oleinik, M. A. Shubin and M. Lesch. We do not assume a priori that M is endowed with a complete Riemannian metric. This allows us to treat e.g. operators acting on bounded domains in R n with the Lebesgue measure. We also allow singular potentials V . In particular, we obtain a new self-adjointness condition for a Schrödinger operator on R n whose potential has the Coulomb-type singularity and is allowed to fall off to −∞ at infinity.For a specific case when the principal symbol of D * D is scalar, we establish more precise results for operators with singular potentials. The proofs are based on an extension of the Kato inequality which modifies and improves a result of