Uniqueness of reconstruction of an \(n\)th-order differential operator with nonseparated boundary conditions by several spectra (Q2243698)

The paper deals with the boundary value problem \(L\) for the sixth-order differential equation \[ y^{(n)}(x) + \sum_{m = 0}^{n-2} p_m(x) y^{(m)}(x) = \lambda y(x), \quad x \in (0, 1), \] with the non-separated boundary conditions \[ \sum_{k = 1}^n a_{ik} y^{(k)}(0) + \sum_{k = n + 1}^{2n} a_{ik} y^{(k)}(1) = 0, \quad i = 1, 2, \dots, n, \] where \(n = 6\), \(p_m\) are complex-valued functions, \(p_m^{(m)} \in L_1(0, 1)\), \(m = 0, 1, \ldots, 4\), \(\lambda\) is the spectral parameter, \(a_{ik}\) are complex coefficients. The authors show that the coefficients of the problem \(L\) can be uniquely reconstructed by using a certain finite number of eigenvalues of \(L\) and \((4n-6)\) spectra of certain auxiliary eigenvalue problems with separated boundary conditions. The proof is based on the results of \textit{E. A. Baranova} [Sov. Math., Dokl. 13, 1095--1098 (1972; Zbl 0271.34029); translation from Dokl. Akad. Nauk SSSR 205, 1271--1273 (1972)].