An inverse spectral problem for a nonnormal first order differential operator
The authors study the eigenvalues of two restrictions of the operator \[ A = \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right) \partial_x + \left( \begin{matrix} r(x) & 0 \\ 0 & g(x) \end{matrix} \right), \] where \(r\) and \(g\) are Lipschitz functions on \([0,1]\). They establish asymptotic formulas for these eigenvalues and show that the corresponding root vectors form a basis in \(L^2 \times L^2\). A related inverse problem was studied by \textit{M. Yamamoto} [J. Fac. Sci. Univ. Tokyo, Sect. IA 35, 519-546 (1988; Zbl 0667.34018)].
scientific article; zbMATH DE number 908002