Journal of Operator Theory

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

Singular spectrum for multidimensional Schrödinger operators with potential barriers

Authors: Peter Stollmann (1) and Günter Stolz (2)
Author institution:(1) Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, D-60054 Frankfurt am Main, Germany
(2) Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, 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 Schrödinger 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.

Keywords: Schrödinger operators, singular spectrum, trace-class scattering.

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