论文标题
关于逆传输特征值问题的正规化方法
On a regularization approach to the inverse transmission eigenvalue problem
论文作者
论文摘要
我们考虑$ -y''+q（x）y =ρ^2 y，$ $ y（0）= y（1）\ cosCTA-y'（1）ρ^{ - 1} \sinρa= 0的不规则（在Birkhoff甚至Stone Sense）的不规则传输特征值问题。全面研究了从传输特征值中恢复潜在的$ q（x）$的独特性问题。在这里，我们研究了这个反问题的可溶性和稳定性。为此，我们建议首先选择所谓的特征值问题的一些常规子类，这实际上决定了研究的过程，甚至决定了逆问题的确切陈述。对于确定性，通过假设$ q（x）$是$ w_2^1 [0,1] $具有零平均值和$ q（1）\ ne0的复杂功能不规则传播特征值问题，还可以从特征性功能方面获得必要和充分的条件，以解决逆问题的可溶性，即从光谱中恢复任意实现的实值的平方累积潜力$ q（x）$，用于任何固定的$ a \ in {\ Mathbb r}中的任何固定$ a \ a
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 $a\in{\mathbb R}.$
