Spectral Multiplicity for Random Operators with Projection valued Randomness

Summary

The main focus of the thesis is to understand the structure of spectrum for a certain class of random operators. The family of operators in consideration can be decomposed into the sum of a fixed self-adjoint operator and sum of a countable family of perturbations by finite rank projections. The perturbations themselves is chosen to be independent real random variables, so the family of random operators is viewed as self-adjoint operator valued random variable.

In the case of l²(Z^d) when the rank of the projections are one and the self-adjoint operator is discrete Laplacian, this family of operator is called the Anderson tight binding model. This was first proposed by P.W. Anderson as a quantum mechanical theory to study spin waves in doped silicon. This model is important for understanding spin-waves and conduction in materials. There are multiple results on its spectrum by various authors, and one of the important results is about simplicity of singular spectrum. Another important model to work on is random Schrodinger operator on the space L²(R^d) where the rank of the projections are infinite. In this case, no one has been able to prove any kind of multiplicity bound.

This work is an intermediate step in understanding the spectral structure of random Schrodinger operator. In this thesis, I study the case when the rank of perturbations are finite and equal. I am able to determine the condition under which the invariant subspace for the operator associated with the projections is unitarily equivalent to each other. Finally, the singular subspace for the operator is given by the singular subspace invariant under action of the operator for any one of the projection. This implication provides a direct way to write the spectral measure for the singular part by looking at the matrix-valued Herglotz function for a single projection, and so gives a bound on multiplicity of the singular spectrum.

Soft copy of the thesis is available here.