Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

Abstract

We study properties of the nodal sets of high frequency eigenfunctions and quasimodes for radial perturbations of the Harmonic Oscillator. In particular, we consider nodal sets on spheres of large radius (in the classically forbidden region) for quasimodes with energies lying in intervals around a fixed energy E. For well chosen intervals we show that these nodal sets exhibit quantitatively different behavior compared to those of the unperturbed Harmonic Oscillator. These energy intervals are defined via a careful analysis of the eigenvalue spacings for the perturbed operator, based on analytic perturbation theory and linearization formulas for Laguerre polynomials.

Sharp endpoint estimates for eigenfunctions restricted to submanifolds of codimension 2

“…The Hermite operator −∆ + |x| 2 in R n shares some similar features with the spherical Laplacian, such as periodic Hamilton flow and many highly concentrated eigenfunctions, and the problem of obtaining L p eigenfunction bounds has received considerable interest in the context of Bochner-Riesz means [41,40,42,28,23,14,13,12,15], as well as unique continuation problems [16,17,30]. To understand the nodal sets of the Hermite eigenfunctions in R n , the sizes of nodal sets in small balls have been studied, see Bérard-Helffer [4,3], Hanin-Zelditch-Zhou [19,20], Beck-Hanin [2] and Jin [27]. In this paper, we investigate the concentration of the Hermite eigenfunctions in R n by establishing sharp L p bounds over balls.…”