Spectral Asymptotics for Large Skew-Symmetric Perturbations of the Harmonic Oscillator

Initially motivated by a problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator H-epsilon = -partial derivative(2)(x) + x(2) + i epsilon(-1) f(x) on L-2(R), where f is a real-valued function and epsilon > 0 is 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) and Psi(epsilon) go to infinity as epsilon -> 0, and we give precise estimates of the growth rate of Psi(epsilon). We also provide an example where Sigma(epsilon) >> Psi(epsilon) if epsilon is small. Our main results are established using variational "hypocoercive" methods, localization techniques, and semiclassical subelliptic estimates.