We discuss the relevance of the classical Riccati substitution to the spectral edge statistics in some fundamental models of one-dimensional random Schrödinger and random matrix theory.

1. Introduction

The Riccati map amounts to the observation that the Schr¨odinger eigenvalue problemis transformed into the first order relation

upon setting. That this simple fact has deep consequences for the problem of characterizing the spectrum of Q with a random potential q has been known for some time. It also turns out to be important for related efforts in random matrix theory (RMT). We will describe some of the recent progress on both fronts.

Random operators of type Q arise in the description of disordered systems. Their use goes back to Schmidt [1957], Lax and Phillips [1958], and Frisch and Lloyd [1960] in connection with disordered crystals, represented by potentials in the form of trains of signed random masses, randomly placed on the line. Consider instead the case of white noise potential, q(x) =b′(x) with a standard brownian motion x ⟼ b(x), which may be viewed as a simplifying caricature of the above.