08-155 I. Gallagher, Th. Gallay, F. Nier

Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator (523K, PDF) Sep 3, 08

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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.

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