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Article Dans Une Revue International Mathematics Research Notices Année : 2009

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

Résumé

Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator H ǫ = −∂ 2 x + x 2 + iǫ −1 f (x) on L 2 (R), where f is a real-valued function and ǫ > 0 a small parameter. We define Σ(ǫ) as the infimum of the real part of the spectrum of H ǫ , and Ψ(ǫ) −1 as the supremum of the norm of the resolvent of H ǫ along the imaginary axis. Under appropriate conditions on f , we show that both quantities Σ(ǫ), Ψ(ǫ) go to infinity as ǫ → 0, and we give precise estimates of the growth rate of Ψ(ǫ). We also provide an example where Σ(ǫ) ≫ Ψ(ǫ) if ǫ is small. Our main results are established using variational " hypocoercive " methods, localization techniques and semiclassical subelliptic estimates.
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Dates et versions

hal-01227470 , version 1 (11-11-2015)

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Isabelle Gallagher, Thierry Gallay, Francis Nier. Spectral Asymptotics for Large Skew-Symmetric Perturbations of the Harmonic Oscillator. International Mathematics Research Notices, 2009, 2009 (12), pp.2147-2199. ⟨10.1093/imrn/rnp013⟩. ⟨hal-01227470⟩
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