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.