arXiv 2005.117332020-05-24math.SP

传输本征值问题的正则化方法

On a regularization approach to the inverse transmission eigenvalue problem

S. A. Buterin, A. E. Choque-Rivero, M. A. Kuznetsova

备注：: 18 pages
DOI：: 10.1088/1361-6420/abaf3c

摘要：

我们考虑的不规则（在伯克霍夫，甚至斯通意义上）传输本征值问题的形式$-y''+q(x)y=ρ^2 y,$ $y(0)=y(1)\cosρa-y'(1)ρ^{-1}\sinρa=0.$的主要重点是“最"的不规则情况下$a=1,$，这是很重要的应用。全面研究了由传输特征值恢复势$q(x)$的唯一性问题。在这里，我们研究这个反问题的可解性和稳定性。为此，我们建议所谓的正则化方法，根据该方法，首先应该选择一些考虑中的特征值问题的正则子类，这实际上决定了研究的过程，甚至是反问题的精确陈述。为明确起见，通过假设$q(x)$是$W_2^1[0,1]$中具有零均值的复值函数和$q(1)\ne0,$，我们研究了传输本征值的性质，并证明了从谱中恢复$q(x)$以及值$q(1).$的局部可解性和稳定性。在附录中，我们提供了一些正则和非正则传输本征值问题的说明性例子，对于任意固定的$a\in{\mathbb R}.$，得到了由谱恢复任意实值平方可积势$q(x)$反问题可解的特征函数的充要条件

英文摘要：

We consider the irregular (in the Birkhoff and even the Stone sense) transmission eigenvalue problem of the form $-y''+q(x)y=ρ^2 y,$ $y(0)=y(1)\cosρa-y'(1)ρ^{-1}\sinρa=0.$ The main focus is on the ''most'' irregular case $a=1,$ which is important for applications. The uniqueness questions of recovering the potential $q(x)$ from transmission eigenvalues were studied comprehensively. Here we investigate the solvability and stability of this inverse problem. For this purpose, we suggest the so-called regularization approach, under which there should first be chosen some regular subclass of eigenvalue problems under consideration, which actually determines the course of the study and even the precise statement of the inverse problem. For definiteness, by assuming $q(x)$ to be a complex-valued function in $W_2^1[0,1]$ possessing the zero mean value and $q(1)\ne0,$ we study properties of transmission eigenvalues and prove local solvability and stability of recovering $q(x)$ from the spectrum along with the value $q(1).$ In Appendices, we provide some illustrative examples of regular and irregular transmission eigenvalue problems, and also obtain necessary and sufficient conditions in terms of the characteristic function for solvability of the inverse problem of recovering an arbitrary real-valued square-integrable potential $q(x)$ from the spectrum, for any fixed $a\in{\mathbb R}.$

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英文摘要：

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