  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.
