Concentration of eigenfunctions of Schroedinger operators

TL;DR

This paper studies the asymptotic distribution of eigenfunctions of single-well Schrödinger operators, revealing how their limit measures depend on the potential's growth rate, with implications for quantum mechanics and spectral theory.

Contribution

It characterizes the limit measures of eigenfunctions for a class of Schrödinger operators with different potential growth behaviors, extending previous understanding in spectral analysis.

Findings

Limit measure supported on [-1,1]

Density proportional to (1-|x|^β)^(-1/2) for polynomial potentials

Uniform density for super-polynomial potentials

Abstract

We consider the limit measures induced by the rescaled eigenfunctions of single-well Schr\"odinger operators. We show that the limit measure is supported on $[-1,1]$ and with the density proportional to $(1-|x|^\beta)^{-1/2}$ when the non-perturbed potential resembles $|x|^\beta$, $\beta >0$, for large $x$, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.