论文标题
具有有限等级扰动的Schrödinger方程的分散估计值
Dispersive estimates for the Schrödinger equation with finite rank perturbations
论文作者
论文摘要
在本文中，我们调查了哈密顿人$ h =-Δ+sum_ { d \ ge 1，$$，每个$φ_j$都满足某些平滑度和衰减条件。我们表明，在频谱假设下，存在常数$ c = c（n，d，φ_1，\ ldots，φ_n）> 0 $，以至于$ \ | e^{ - ith} \ | ___ {l^1-l^1-l^1-l^{\ infty}}} \，\，\，\ text {for} \，\，\，t> 0。 $$ 据我们所知，这似乎提供了第一次研究$ l^1-l^{\ infty} $估计值，以在任何维度上对Laplacian的有限等级扰动。 我们首先处理等级的一个扰动（$ n = 1 $）。然后，我们转向一般情况。我们方法中的新想法是为有限等级扰动建立Aronszajn-Krein类型公式。这使我们能够将分析减少到等级的情况下，并以统一的方式解决问题。此外，我们表明，在某些特定情况下，常数$ c（n，d，φ_1，\ ldots，φ_n）$在$ n $中多样地生长。最后，作为一个应用程序，我们能够将结果扩展到$ n = \ infty $并处理一些跟踪类扰动。
In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$ H=-Δ+\sum_{j=1}^N\langle\cdot\,, φ_j\rangle φ_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1, $$ where each $φ_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, φ_1,\ldots, φ_N)>0$ such that $$ \|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0. $$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ estimates for finite rank perturbations of the Laplacian in any dimension. We first deal with rank one perturbations ($N=1$). Then we turn to the general case. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant $C(N, d, φ_1,\ldots, φ_N)$ grows polynomially in $N$. Finally, as an application, we are able to extend the results to $N=\infty$ and deal with some trace class perturbations.
