Smoothness of the Integrated Density of States for Random Schrödinger Operators on Multidimensional Strips

Citation

Glaffig, Clemens H. (1988) Smoothness of the Integrated Density of States for Random Schrödinger Operators on Multidimensional Strips. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4GBV-RA25. https://resolver.caltech.edu/CaltechETD:etd-09012005-155238

Abstract

We investigate smoothness properties of the integrated density of states (ids) for random Schrödinger operators on a multidimensional strip lattice, where only the potentials on the "top surface" of this lattice have a distribution with some regularity.

We view the eigenvalue equation on the strip as the action of an abstract group on some homogeneous space, from where we derive a representation of the ids in terms of a distinguished measure on that homogeneous space.

This representation allows us to conclude that using minimal smoothness of the potential distribution on the "top surface", combined with a negative moment condition for the distribution of all other potentials, is enough to obtain smoothness of the ids. This includes the original Anderson model.

We also discuss cases, where the distribution of the potentials below the "top surface" is Bernoulli, satisfying this negative moment condition.

Thesis Files