Spectral theory and resonance for Schrodinger operators

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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.