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.
| Origine | Fichiers produits par l'(les) auteur(s) |
|---|
Loading...