Christian Remling
Some Schr\"odinger operators with power-decaying potentials and
pure point spectrum
(38K, LaTeX)
ABSTRACT. We construct deterministic (= non-random) potentials $V(x)=O(x^{-c})$
such that the one-dimensional Schr\"odinger equation $-y''+Vy=Ey$
on the half-axis $x\in [0,\infty)$ has dense pure point spectrum
in $(0,\infty)$ for almost all boundary conditions at $x=0$.
A modification of this construction yields power-decaying potentials
for which the spectrum is purely singular continuous in $(0,\infty)$
for all boundary conditions.