A universality law for sign correlations of eigenfunctions of differential operators

Abstract

We establish a sign correlation universality law for sequences of functions $\{w_n{\}}_{n\in \mathbb{N}}$ satisfying a trigonometric asymptotic law. Our results are inspired by the classical WKB asymptotic approximation for Sturm–Liouville operators, and in particular we obtain non-trivial sign correlations for eigenfunctions of generic Schrödinger operators (including the harmonic oscillator), as well as Laguerre and Chebyshev polynomials. Given two distinct points $x,y\in \mathbb{R}$, we ask how often do $w_n(x)$ and $w_n(y)$ have the same sign. Asymptotically, one would expect this to be true half the time, but this turns out to not always be the case. Under certain natural assumptions, we prove that, for all $x\ne y$,

and that these bounds are optimal, and can be attained. Our methods extend to other problems of similar flavor and we also discuss a number of open problems.