G\"unter Stolz
Localization for Schr\"odinger operators with effective barriers
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ABSTRACT. We consider two Schr\"odinger operators $H_0 = -\Delta +V_0$ and $H=-\Delta
+V$ in $L^2(\mbox{\bf R}^d)$, the dimension $d$ being arbitrary. We assume
that the resolvent set $\rho(H_0)$ of $H_0$ is known and that the potentials
$V_0(x)$ and $V(x)$ coincide on a sequence of ring shaped (but not
necessarily spherical) regions $U_n$, $n=1,2,\ldots$. Under certain
assumptions on the geometry of the $U_n$ and their complements we prove the
following result for the operator family $H(\lambda) = H+\lambda W$, where
$W$ is a continuous and compactly supported potential of fixed sign and
$\sigma_{ac}$ and $\sigma_c$ denote the absolutely continuous and continuous
spectrum, respectively: (i) $\sigma_{ac}(H(\lambda)) \cap \rho(H_0) =
\emptyset$ for every $\lambda \in \mbox{\bf R}$, (ii) $\sigma_c(H(\lambda))
\cap \rho(H_0) = \emptyset$ for almost every $\lambda \in \mbox{\bf R}$ with
respect to Lebesgue measure. We also give an explicit decay estimate for
eigenfunctions, thus establishing localization for $H(\lambda)$ in arbitrary
spectral gaps of $H_0$. The result can be applied to potentials with a
sequence of wide barriers as well as to perturbations of periodic potentials.