谐振子反对称大扰动的谱渐近性
Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator
摘要：
最初的动机是在流体力学中的稳定性问题，我们研究的频谱和伪谱性质的微分算子$H_ε= -\partial_x^2 + x^2 + iε^{-1}f(x)$上的$L^2(R)$，其中$f$是一个实值函数和$ε> 0$的小参数。定义$Σ(ε)$为$H_ε$的谱的实部的下确界，$Ψ(ε)^{-1}$为沿虚轴的$H_ε$的预解式的范数的上确界.在适当的条件下，我们证明了$f$上的两个量$Σ(ε)$，$Ψ(ε)$都以$ε\to 0$的形式趋于无穷大，并给出了$Ψ(ε)$的增长率的精确估计.我们还提供了一个例子，其中如果$ε$很小，则$Σ(ε)$比$Ψ(ε)$大得多。我们的主要结果建立使用变分“hypocotive”的方法，本地化技术和半经典次椭圆估计。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
