We study two inverse problems of recovering the complex-valued squareintegrable potential q(x) in the operator −y ′′ (x) + q(x)y(a), y (ν) (0) = γy (ν) (1) for ν = 0, 1, where γ ∈ C \ {0}. The first problem involves only single spectrum as the input data. We obtain complete characterization of the spectrum and prove that its specification determines q(x) uniquely if and only if γ = ±1. For the rest cases: periodic and antiperiodic ones, we describe classes of iso-spectral potentials and provide restrictions under which the uniqueness holds. The second inverse problem deals with recovering q(x) from the two spectra related to γ = ±1. We obtain necessary and sufficient conditions for its solvability and establish that uniqueness holds if and only if a = 0, 1. For a ∈ (0, 1), we describe classes of iso-bispectral potentials and provide restrictions under which the uniqueness resumes. Algorithms for solving both inverse problems are provided.

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In the paper, Sturm-Liouville differential operators on time scales consisting of a finite number of isolated points and segments are considered. Such operators unify differential and difference operators. We obtain properties of their spectral characteristics including asymptotic formulae for eigenvalues and weight numbers. Uniqueness theorem is proved for recovering the operators from the spectral characteristics.

We consider the irregular (in the Birkhoff and even the Stone sense) transmission eigenvalue problem of the form −y″ + q(x)y = ρ 2 y, y(0) = y(1) cos ρa − y′(1)ρ −1 sin ρa = 0. The main focus is on the ‘most’ irregular case a = 1, which is important for applications. The uniqueness questions of recovering the potential q(x) from transmission eigenvalues were studied comprehensively. Here we investigate the solvability and stability of this inverse problem. For this purpose, we suggest the so-called regularization approach, under which there should first be chosen some regular subclass of eigenvalue problems under consideration, which actually determines the course of the study and even the precise statement of the inverse problem. For definiteness, by assuming q(x) to be a complex-valued function in W 2 1 [ 0 , 1 ] possessing the zero mean value and q(1) ≠ 0, we study properties of transmission eigenvalues and prove the local solvability and stability of recovering q(x) from the spectrum along with the value q(1). In the appendices, we provide some illustrative examples of regular and irregular transmission eigenvalue problems, and we also obtain necessary and sufficient conditions in terms of the characteristic function for the solvability of the inverse problem of recovering an arbitrary real-valued square-integrable potential q(x) from the spectrum for any fixed a ∈ R .

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