Inverse transmission eigenvalue problems for the Schrödinger operator with the Robin boundary condition (Q2667941)

The paper deals with the following transmission eigenvalue problem \(H(q,h)\) \begin{align*} -\varphi^{\prime\prime}+q(x)\varphi=k^{2}\varphi,\qquad x\in(0,1),\\ -\varphi_{0}^{\prime\prime}=k^{2}\varphi_{0},\qquad x\in(0,1),\\ \varphi^{\prime}(0)+h\varphi(0)=0,\\ \varphi_{0}^{\prime}(0)+h_{0}\varphi_{0}(0)=0,\\ \varphi(1)=\varphi_{0}(1),\,\,\varphi^{\prime}(1)=\varphi_{0}^{\prime}(1). \end{align*} Here \(k\) is the spectral parameter, \(q\) is the complex-valued potential \(q\in L_{2}(0,1)\), \(h_{0},h\in \mathbf{C}.\) By considering the distinctness of the parameters \(h_{0},h\), the authors show that the potential \(q(x)\) and the parameter \(h\) can be uniquely determined by all transmission eigenvalues and a certain constant on the condition that \(h_{0}\) is known. This partially solves the open question raised by [\textit{T. Aktosun} and \textit{V. G. Papanicolaou}, Inverse Probl. 30, No. 7, Article ID 075001, 23 p. (2014; Zbl 1305.34145)]. Moreover, the authors prove the local solvability and stability of recovering \(h\) and \(q(x)\) in a particular set from the spectrum and the non-zero value \(q(1)\).