We consider the irregular (in the Birkhoff and even the Stone sense) transmission eigenvalue problem of the form $-y''+q(x)y=\rho^2 y,$ $y(0)=y(1)\cos\rho a-y'(1)\rho^{-1}\sin\rho a=0.$ 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 $a\in{\mathbb R}.$

中文翻译：

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逆传输特征值问题的正则化方法

我们考虑形式为 $-y''+q(x)y=\rho^2 y,$ $y(0)=y(1)\ cos\rho a-y'(1)\rho^{-1}\sin\rho a=0.$ 主要关注“最”不规则情况 $a=1,$ 这对应用程序很重要. 全面研究了从传输特征值中恢复潜在$q(x)$的唯一性问题。在这里，我们研究了这个逆问题的可解性和稳定性。为此，我们建议采用所谓的正则化方法，在这种方法下，首先应选择一些正在考虑的特征值问题的正则子类，这实际上决定了研究的过程，甚至决定了逆问题的精确表述。为了确定性，