Regularity and decay of solutions of nonlinear harmonic oscillators

TL;DR

This paper establishes sharp regularity and decay properties of solutions to nonlinear harmonic oscillators with variable coefficients, demonstrating holomorphic extension and Gaussian decay based on Hermite function properties.

Contribution

It provides new results on the analytic regularity and decay at infinity for solutions of variable coefficient nonlinear harmonic oscillators, including applications to nonlinear eigenvalue problems.

Findings

Solutions extend holomorphically to a sector in the complex domain.

Solutions exhibit Gaussian decay at infinity.

Results apply to nonlinear eigenvalue problems with real-analytic scattering metrics.

Abstract

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in R^d. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in R^d, as well as to certain perturbations of the classical harmonic oscillator.