01-387 Rowan Killip

Perturbations of One-Dimensional Schr\"odinger Operators Preserving the Absolutely Continuous Spectrum (83K, AMS LaTeX) Oct 19, 01

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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.

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