Journal of Operator Theory

Volume 32, Issue 1, Summer 1994 pp. 91-109.

Singular spectrum for multidimensional Schrodinger operators with potential barriers

Authors: Peter Stollmann (1) and Guenter Stolz (2)
Author institution: (1) Fachbereich Mathematik, Johann Wolfgang Goethe-Universitaet, D-60054 Frankfurt am Main, Germany
(2) Fachbereich Mathematik, Johann Wolfgang Goethe-Universitaet, D-60054 Frankfurt am Main, Germany and University of Alabama, Dept. of Math., Birmingham, Alabama 35294, U.S.A.

Summary: We prove comparison criteria for the absence of absolutely continuous spectrum which have the following form: If the set

$\{x; V_0 (x) \ne V(x)\}$ can be divided into bounded parts with suitable geometric conditions, then

$\sigma _{{\rm{ac}}} ( - \frac{1}{2}\Delta + V) \subset [\inf {\rm{ }}\sigma _{{\rm{ess}}} ( - \frac{1}{2}\Delta + V_0 ), {\rm{ }}\infty )$ , or, under somewhat stronger conditions,

$\sigma _{{\rm{ac}}} ( - \frac{1}{2}\Delta + V) \subset \sigma _{{\rm{ess}}} ( - \frac{1}{2}\Delta + V_0 ),{\rm{ }}\infty )$ . The first result proves absence of absolute continuity for

$- \frac{1}{2}\Delta + V$ below

$\inf {\rm{ }}\sigma _{{\rm{ess}}} ( - \frac{1}{2}\Delta + V_0 )$ and is a continuum analog of a result for discrete Schroedinger operators by Simon and Spencer. The second inclusion implies in addition that

$- \frac{1}{2}\Delta + V$ has no absolutely continuous spectrum in arbitrary gaps of

$\sigma _{{\rm{ess}}} ( - \frac{1}{2}\Delta + V_0 )$ . One should think of applying this to a given V by constructing V_0 suitably in order to produce prescribed gaps in

$\sigma _{{\rm{ess}}} ( - \frac{1}{2}\Delta + V_0 )$ . Different potentials V_0 may be associated with one and the same V in order to exclude absolute continuity in varying intervals.

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