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 $H_ε= -\partial_x^2 + x^2 + iε^{-1}f(x)$ on $L^2(R)$, where $f$ is a real-valued function and $ε> 0$ a small parameter. We define $Σ(ε)$ as the infimum of the real part of the spectrum of $H_ε$, and $Ψ(ε)^{-1}$ as the supremum of the norm of the resolvent of $H_ε$ along the imaginary axis. Under appropriate conditions on $f$, we show that both quantities $Σ(ε)$, $Ψ(ε)$ go to infinity as $ε\to 0$, and we give precise estimates of the growth rate of $Ψ(ε)$. We also provide an example where $Σ(ε)$ is much larger than $Ψ(ε)$ if $ε$ is small. Our main results are established using variational "hypocoercive" methods, localization techniques and semiclassical subelliptic estimates.