00-326 Rowan Killip
Perturbations of One-Dimensional Schr\"odinger Operators Preserving the Absolutely Continuous Spectrum (107K, AMS-LaTeX) Aug 28, 00
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Abstract. We study the stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators $$[Hu](x) = -u''(x) + q(x)u(x)$$ with periodic potentials $q(x)$. Specifically, it is proved that any perturbation of the potential, $V\in L^2$, preserves the essential support (and multiplicity) of the absolutely continuous spectrum. This is optimal in terms of $L^p$ spaces and, for $q\equiv 0$, it answers in the affirmative a conjecture of Kiselev, Last and Simon. By adding constraints on the Fourier transform of $V$, it is possible to relax the decay assumptions on $V$. It is proved that if $V\in L^3$ and $\hat V$ is uniformly locally square integrable, then preservation of the a.c.~spectrum still holds. If we assume that $q\equiv0$, still stronger results follow: if $V\in L^3$ and $\hat V(k)$ is square integrable on an interval $[k_0,k_1]$, then the interval $[k_0^2/4,k_1^2/4]$ is contained in the essential support of the absolutely continuous spectrum of the perturbed operator.

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