Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator

TL;DR

This paper investigates the spectral and pseudospectral behavior of a complex perturbed harmonic oscillator operator, revealing how the spectrum's real part and resolvent norms grow as the perturbation parameter diminishes.

Contribution

It provides new precise estimates on the growth rates of spectral and pseudospectral quantities for large skew-symmetric perturbations of the harmonic oscillator, using hypocoercive and semiclassical methods.

Findings

Both the spectrum's real part and the resolvent norm tend to infinity as the perturbation vanishes.

The growth rates of these quantities are explicitly estimated.

An example demonstrates the spectrum can grow faster than the pseudospectrum in certain cases.

Abstract

Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator $H_\epsilon = -\partial_x^2 + x^2 + i\epsilon^{-1}f(x)$ on $L^2(R)$, where $f$ is a real-valued function and $\epsilon > 0$ a small parameter. We define $\Sigma(\epsilon)$ as the infimum of the real part of the spectrum of $H_\epsilon$, and $\Psi(\epsilon)^{-1}$ as the supremum of the norm of the resolvent of $H_\epsilon$ along the imaginary axis. Under appropriate conditions on $f$, we show that both quantities $\Sigma(\epsilon)$, $\Psi(\epsilon)$ go to infinity as $\epsilon \to 0$, and we give precise estimates of the growth rate of $\Psi(\epsilon)$. We also provide an example where $\Sigma(\epsilon)$ is much larger than $\Psi(\epsilon)$ if $\epsilon$ is small. Our main results are established using variational "hypocoercive" methods, localization techniques and semiclassical subelliptic estimates.