Semigroups for One-Dimensional Schrödinger Operators with Multiplicative Gaussian Noise

Abstract

A problem of fundamental interest in mathematical physics is that of understanding the structure of the spectrum of random Schrödinger operators. An important tool in such investigations is the Feynman-Kac formula, which provides a simple probabilistic representation of the semigroup of Schrödinger operators, thus making exponential functionals of the eigenvalues amenable to explicit computation. Classical results in the theory of random Schrödinger semigroups concern discrete operators (i.e., acting on a lattice) or continuous operators with a smooth random noise. In many applications, however, it is more natural to consider continuous operators with very irregular random noises modelled by Schwartz distributions (such as Gaussian white noise).

In this thesis, we take the first steps in developing a general semigroup theory for one-dimensional random Schrödinger operators whose noise is given by the formal derivative of a continuous Gaussian process with stationary increments. The main source of inspiration for these results is the stochastic semigroup theory pioneered by Gorin and Shkolnikov in the random matrix theory literature. Our main result consists of a Feynman-Kac formula for such operators, which naturally extends the classical Feynman-Kac formula for random Schrödinger operators with smooth Gaussian noise. As a consequence of our main result, we obtain an explicit representation of the Laplace transforms of the eigenvalue point process of one-dimensional Schrödinger operators with generalized Gaussian noise in terms of exponential functionals of Brownian local time.

We present two applications of this new semigroup theory. Firstly, we use our new Feynman-Kac formulas to provide the first method capable of proving the occurrence of a property called number rigidity in the spectrum of general random Schrödinger operators with irregular noise. Secondly, we study the convergence of discrete approximations of our Feynman-Kac formulas and their applications in proving limit laws for the extremal eigenvalues of random matrices.