Inverse nodal and inverse spectral problems are studied for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.

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We study the inverse problem for non-selfadjoint Sturm-Liouville operators on a finite interval with possibly multiple spectra. We prove the uniqueness theorem and obtain constructive procedures for solving the inverse problem along with the necessary and sufficient conditions of its solvability and also prove the stability of the solution.

Let P (x) denote a 2 × 2 symmetric matrix-valued function defined on [0, 1]. We prove that if there exists an infinite sequence {y(x; λ n j )} ∞ j=1 of Dirichlet eigenfunctions of the operator − d 2 dx 2 + P (x) whose components all have zeros in common, then P (x) is simultaneously diagonalizable on [0,1]. This result can also be generalized to the general n-dimensional case.