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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In their book The Markov Moment Problem and Extremal Problems, published in 1977, M. G. Krein and A. A. Nudel'man presented a complete solution of the truncated Hausdorff moment problem via orthogonal polynomials on a finite interval [a, b]. By using the Potapov schema the matrix version of this moment problem was studied by the author, Yu. M. Dyukarev, B. Fritzsche and B. Kirstein.In the present work, we obtain the matrix generalisation of the above-mentioned Krein-Nudel'man representation. We also obtain explicit relations between four families of orthogonal matrix polynomials on [a, b] and their second kind polynomials, which are associated with the matrix version of the truncated Hausdorff moment problem. Mathematics Subject Classification

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