Spectral theory and resonance for Schrodinger operators

我的研究是关于薛定谔算子的谱和散射理论，以及偏微分方程和几何的相关主题。近年来，我一直致力于证明在无穷远处行为不规则的离散薛定谔算子的绝对连续谱的存在性。这类算子的主要例子是具有随机势的薛定谔算子。 这项工作的动机之一是扩展态猜想，这是随机薛定谔算子领域中一个尚未解决的大问题。这个猜想断言，对于三维或更高维整数格点上的安德森模型，在低无序度下存在一些绝对连续的谱。这个猜想的物理意义是，如果无序度足够低，无序固体应该具有一些导电能。唯一的情况下，这个猜想已被证明是当整数格被替换为一棵树。最初的证明是由于阿贝尔克莱因。在我与Hasler和Spitzer的工作中，我发现了这个结果的一个新证明，并建议使用这个证明中的思想来研究一系列问题，这些问题与整数格的工作有一些共同的困难。特别是，我现在能够处理一些具有无限长度循环的图。我还计划联合收割机我们的方法与运营商理论技术研究运营商的晶格缓慢衰减的随机潜力。 我对一些涉及共振或散射极点的问题感兴趣。我建议研究一个问题，涉及的位置的共振和一个涉及的长期行为的共振状态。

My research is about spectral and scattering theory of Schrödinger Operators, and related topics in PDE and Geometry. In recent years I have focused on proving the existence of absolutely continuous spectrum for discrete Schrödinger operators whose behavior at infinity is irregular. The primary examples of such operators are Schrödinger operators with random potentials. One motivation for this work is the extended states conjecture, a big unsolved problem in the field of random Schrödinger operators. This conjecture asserts for the Anderson model on the integer lattice in dimensions three or higher there is some absolutely continuous spectrum at low disorder. The physical meaning of this conjecture is that a disordered solid should have some conducting energies provided the disorder is sufficiently low. The only situation where this conjecture has been proved is when the integer lattice is replaced with a tree. The original proof is due to Abel Klein. In my work with Hasler and Spitzer, I found a new proof of this result, and am proposing to use ideas from this proof to study a series of problems that share some of the difficulties of working on the integer lattice. In particular, I am now in a position to handle some graphs with loops of unbounded length. I also plan to combine our method with operator theory techniques to study operators on the lattice with slowly decaying random potentials. I am interested in some problems involving resonances, or scattering poles. I am proposing to study a problem involving the positions of resonances and one involving the long time behavior of resonant states.