International Mathematics Research Notices Advance Access originally published online on February 25, 2009
International Mathematics Research Notices (2009) 2009:2147-2199, doi:10.1093/imrn/rnp013 published on June 16, 2009
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Spectral Asymptotics for Large Skew-Symmetric Perturbations of the Harmonic Oscillator
1 Institut de Mathématiques de Jussieu, Université de Paris 7, Case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France
2 Institut Fourier, Université de Grenoble I, BP 74, 38402 Saint-Martin-d'Hères, France
3 IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
Correspondence: Correspondence to be sent to: thierry.gallay{at}ujf-grenoble.fr
Initially motivated by a problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator 
on 
, where f is a real-valued function and 
is a small parameter. We define 
as the infimum of the real part of the spectrum of 
, and 
as the supremum of the norm of the resolvent of 
along the imaginary axis. Under appropriate conditions on f, we show that both quantities 
and 
go to infinity as 
, and we give precise estimates of the growth rate of 
. We also provide an example where 
if 
is small. Our main results are established using variational "hypocoercive" methods, localization techniques, and semiclassical subelliptic estimates.