Schr"odinger operators on a half-line with inverse square potentials
Schr"odinger operators on a half-line with inverse square potentials
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平方势反比半线上的 Schr"odinger 算子
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
F. Truc
中科院分区：
文献类型：
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作者：
H. Kovařík;F. Truc
我们考虑由式（1.1）给出的Schr^odinger算子$H_alpha$。我们研究了谱密度$E(H_alpha, lambda)$在$lambda$趋于$0$时的渐近行为，以及演化算子$E ^{-i t H_alpha}$所对应的$L^1到$L^ inty $色散估计。特别地，我们证明了当$alpha$为正值时，与具有短程势$V$的Schr ' odinger算子的谱密度相比，$ λ o 0$的谱密度以更快的速度趋于零。然后我们展示了$e^{-i t H_alpha}$的长时间行为是如何依赖于$alpha$的。更准确地说，我们证明了$e^{-i t H_alpha}$对于$t oinfty$的衰减率可以任意大，只要我们选择足够大的$alpha$并考虑合适的算子范数。
We consider Schr^odinger operators $H_alpha$ given by equation (1.1) below. We study the asymptotic behavior of the spectral density $E(H_alpha, lambda)$ when $lambda$ goes to $0$ and the $L^1 o L^infty$ dispersive estimates associated to the evolution operator $e^{-i t H_alpha}$. In particular we prove that for positive values of $alpha$, the spectral density tends to zero as $lambda o 0$ with higher speed compared to the spectral density of Schr"odinger operators with a short-range potential $V$. We then show how the long time behavior of $e^{-i t H_alpha}$ depends on $alpha$. More precisely we show that the decay rate of $e^{-i t H_alpha}$ for $t oinfty$ can be made arbitrarily large provided we choose $alpha$ large enough and consider a suitable operator norm.