I. Gallagher, Th. Gallay, F. Nier
Spectral asymptotics for large skew-symmetric perturbations 
of the harmonic oscillator
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ABSTRACT.  Originally motivated by a stability problem in Fluid Mechanics, 
we study the spectral and pseudospectral properties of the 
differential operator $H_\epsilon = -\partial_x^2 + x^2 + 
if(x)/\epsilon$ 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.