In this talk, we are interested in the semigroup theory of continuous one-dimensional random Schrödinger Operators with white noise. We will begin with a brief reminder of the rigorous definition of these operators as well as some of the problems in which they naturally arise. Then, we will discuss the proof of a Feynman-Kac formula describing their semigroups. In closing, we will showcase an application of this new semigroup theory to the study of rigidity (in the sense of Ghosh-Peres) of random Schrödinger eigenvalue point processes.

Some of the results discussed in this talk are joint work with Promit Ghosal (Columbia) and Yuchen Liao (Michigan).