Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator
| dc.creator | Gallagher, I. | |
| dc.creator | Gallay, Th. | |
| dc.creator | Nier, F. | |
| dc.date | 2008-09-03 | |
| dc.date.accessioned | 2026-07-07T10:00:17Z | |
| dc.date.available | 2026-07-07T10:00:17Z | |
| dc.description | Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator $H_ε= -\partial_x^2 + x^2 + iε^{-1}f(x)$ on $L^2(R)$, where $f$ is a real-valued function and $ε> 0$ a small parameter. We define $Σ(ε)$ as the infimum of the real part of the spectrum of $H_ε$, and $Ψ(ε)^{-1}$ as the supremum of the norm of the resolvent of $H_ε$ along the imaginary axis. Under appropriate conditions on $f$, we show that both quantities $Σ(ε)$, $Ψ(ε)$ go to infinity as $ε\to 0$, and we give precise estimates of the growth rate of $Ψ(ε)$. We also provide an example where $Σ(ε)$ is much larger than $Ψ(ε)$ if $ε$ is small. Our main results are established using variational "hypocoercive" methods, localization techniques and semiclassical subelliptic estimates. | |
| dc.description | 38 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/0809.0574 | |
| dc.identifier | http://arxiv.org/abs/0809.0574 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/168289 | |
| dc.subject | Spectral Theory | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35P15; 35P20; 35P99 | |
| dc.title | Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator | |
| dc.type | text |