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

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.

38 pages, 4 figures