Inverse problem for a fourth-order differential operator with nonseparated boundary conditions (Q1734714)

Boundary value problems are considered for the differential equation \[ y^{(4)}+p_2(x)y''+p_1(x)y'+p_0(x)y=\lambda y,\; x\in (0,1) \] with nonseparated boundary conditions. The authors study the inverse problem of recovering coefficients of the boundary conditions from the given eigenvalues provided that the functions \(p_0(x)\), \(p_1(x)\) and \(p_2(x)\) are known a priori. Uniqueness theorems are formulated for this class of inverse problems. Regrettably, these results are wrong. The reason is that the authors use incorrectly Leibenson's uniqueness result, because the specification of the considered spectral data does not uniquely determines the operator. The correct statement and solution of the inverse spectral problem for arbitrary order differential operators are presented in [\textit{V. Yurko}, Method of spectral mappings in the inverse problem theory. Utrecht: VSP (2002; Zbl 1098.34008)].