On the missing eigenvalue problem for Dirac operators
The authors study the Dirac system in the form \[ \begin{cases} u_2' + V_1(x) u_1 = \lambda u_1, \\ - u_1' + V_2(x) u_2 = \lambda u_2, \end{cases} \quad x \in (0, \pi), \] with the boundary conditions \[ \begin{aligned} u_1(0, \lambda) \sin \alpha + u_2(0, \lambda) \cos \alpha = 0, \\ u_1(\pi, \lambda) \sin \beta + u_2(\pi, \lambda) \cos \beta = 0, \end{aligned} \] where \(0 \leq \alpha, \beta < \pi\), \(V_1\) and \(V_2\) are absolutely continuous real-valued functions. Two inverse problems are investigated: the Borg-type problem consists in recovering the potentials \(V_1\) and \(V_2\) from two spectra (the second spectrum corresponds to the boundary conditions with \(\beta_0\) instead of \(\beta\), \(\beta \neq \beta_0\)), and the Hochstadt-Lieberman-type problem consists in recovering \(V_1\) and \(V_2\) from one spectrum under the assumption that \(V_1\) and \(V_2\) are known a priori on \([\pi/2, \pi]\). In both cases, the coefficients \(\alpha\) and \(\beta\) are assumed to be known a priori. The authors prove that the potentials are uniquely defined by two spectra with a missing eigenvalue in the Borg case, and by one spectrum with a missing eigenvalue in the Hochstadt-Lieberman case.
scientific article; zbMATH DE number 6867975
scientific article