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.