Sonin's argument, the shape of solitons, and the most stably singular matrix
Sonin's argument, the shape of solitons, and the most stably singular matrix
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发表时间：
2018-11
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通讯作者：
R. Killip;M. Vişan
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作者：
R. Killip;M. Vişan
我们提出了两个适应的参数Sonin，这是已知的是一个强大的工具，用于获得定性和定量的信息，特殊的功能。我们的具体应用如下：（i）我们给出了一个严格的公式和证明以下断言在任何维度上聚焦NLS：在排斥势球对称孤子的空间包络是半径的非增函数。(ii)受确定最稳定奇异矩阵问题的驱动，我们确定了$n\times n$ GUE矩阵的最大特征值密度的位置。引人注目的是，在偶数维中，这个最大值不是零。
We present two adaptations of an argument of Sonin, which is known to be a powerful tool for obtaining both qualitative and quantitative information about special functions. Our particular applications are as follows: (i) We give a rigorous formulation and proof of the following assertion about focusing NLS in any dimension: The spatial envelope of a spherically symmetric soliton in a repulsive potential is a non-increasing function of the radius. (ii) Driven by the question of determining the most stably singular matrix, we determine the location of the maximal eigenvalue density of an $n\times n$ GUE matrix. Strikingly, in even dimensions, this maximum is \emph{not} at zero.