Rowan Killip
Perturbations of One-Dimensional Schr\"odinger Operators Preserving the Absolutely Continuous Spectrum
(83K, AMS LaTeX)
ABSTRACT. The stability of the absolutely continuous spectrum of
one-di\-men\-sion\-al Schr\"o\-dinger operators,
$$
[Hu](x) = -u''(x) + q(x)u(x),
$$
under perturbations of the potential is discussed. The focus
is on demonstrating this stability under minimal assumptions
on how fast the perturbation decays at infinity.
A general technique is presented together with sample applications.
These include the following: for an operator with a periodic potential,
any perturbation $V\in L^2$ preserves the a.c.spectrum.
For the Stark operator, the same is true for pertubations with
$\int |V(t^2)|^2\, dt <\infty$.
Both of these results are known to be optimal, in the sense that the
integrability index cannot be increased.