197 research outputs found
Essential self-adjointness for semi-bounded magnetic Schr\"odinger operators on non-compact manifolds
We prove essential self-adjointness for semi-bounded below magnetic
Schr\"odinger operators on complete Riemannian manifolds with a given positive
smooth measure which is fixed independently of the metric. Some singularities
of the scalar potential are allowed.
This is an extension of the Povzner--Wienholtz--Simader theorem. The proof
uses the scheme of Wienholtz but requires a refined invariant integration by
parts technique, as well as a use of a family of cut-off functions which are
constructed by a non-trivial smoothing procedure due to Karcher.Comment: 24 pages, revised version, to appear in Journal of Functional
Analysi
Discreteness of spectrum and positivity criteria for Schr\"odinger operators
We provide a class of necessary and sufficient conditions for the
discreteness of spectrum of Schr\"odinger operators with scalar potentials
which are semibounded below. The classical discreteness of spectrum criterion
by A.M.Molchanov (1953) uses a notion of negligible set in a cube as a set
whose Wiener's capacity is less than a small constant times the capacity of the
cube. We prove that this constant can be taken arbitrarily between 0 and 1.
This solves a problem formulated by I.M.Gelfand in 1953. Moreover, we extend
the notion of negligibility by allowing the constant to depend on the size of
the cube. We give a complete description of all negligibility conditions of
this kind. The a priori equivalence of our conditions involving different
negligibility classes is a non-trivial property of the capacity. We also
establish similar strict positivity criteria for the Schr\"odinger operators
with non-negative potentials.Comment: 24 pages, final version, some minor misprints correcte
Essential self-adjointness of Schroedinger type operators on manifolds
We obtain several essential self-adjointness conditions for a Schroedinger
type operator D*D+V acting in sections of a vector bundle over a manifold M.
Here V is a locally square-integrable bundle map. Our conditions are expressed
in terms of completeness of certain metrics on M; these metrics are naturally
associated to the operator. 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 the euclidean space.
For the case when the principal symbol of the operator is scalar, we
establish more precise results. The proofs are based on an extension of the
Kato inequality which modifies and improves a result of Hess, Schrader and
Uhlenbrock.Comment: 52 pages, Minor corrections are made; To appear in Russian Math.
Survey
The Miura Map on the Line
The Miura map (introduced by Miura) is a nonlinear map between function
spaces which transforms smooth solutions of the modified Korteweg - de Vries
equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this
paper we study relations between the Miura map and Schroedinger operators with
real-valued distributional potentials (possibly not decaying at infinity) from
various spaces. We also investigate mapping properties of the Miura map in
these spaces. As an application we prove existence of solutions of the Korteweg
- de Vries equation in the negative Sobolev space H^{-1} for the initial data
in the range of the Miura map.Comment: 33 page
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