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

TL;DR

This paper investigates how radial perturbations affect the high-frequency eigenfunctions' nodal sets of the harmonic oscillator, revealing significant differences in their structure at large radii compared to the unperturbed case.

Contribution

It introduces a detailed analysis of eigenvalue spacings and nodal set behavior under radial perturbations, using analytic perturbation theory and Laguerre polynomial linearization.

Findings

Nodal sets on large spheres differ significantly from the unperturbed case.

Eigenvalue spacings are analyzed to identify energy intervals with distinct nodal behavior.

Perturbations cause quantitative changes in the structure of high-frequency eigenfunctions.

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.