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spectral mappings. A constructive algorithm for the solution of the inverse problem is provided. Keywords:Matrix quadratic dierential pencils, spectral data, inverse spectral problems, method of spectral mappings MSC 2010: 34A55, 34B07, 34B24, 34L40, 47E05 || Natalia Bondarenko: Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia, e-mail: bondarenkonp@info.sgu.ru 1 Introduction In this paper, we consider the boundary value problem L = L(ℓ, U, V) for the equation ℓY := Y + (ñ2I + 2iñQ1(x) + Q0(x))Y = 0, x ∈ (0, ð

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In this work we investigate the completeness, minimality and basis properties of the eigenfunctions of one class discontinuous Sturm-Liouville equation with a spectral parameter in boundary conditions.

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We present an approximation method for Picard second order boundary value problems with Carathéodory righthand side. The method is based on the idea of replacing a measurable function in the right-hand side of the problem with its Kantorovich polynomial. We will show that this approximation scheme recovers essential solutions to the original BVP. We also consider the corresponding finite dimensional problem. We suggest a suitable mapping of solutions to finite dimensional problems to piecewise constant functions so that the later approximate a solution to the original BVP. That is why the presented idea may be used in numerical computations.

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We give a short review of results on inverse spectral problems for second-order differential operators on an interval with non-separated boundary conditions. We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics. In the first part of the review, we provide the main results and methods related to inverse problems for Sturm–Liouville operators with non-separated boundary conditions: periodic, quasi-periodic and Robin-type boundary conditions. At the end, we present the main results on inverse problems for differential pencils with non-separated boundary conditions.

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In this work, we consider inverse nodal problems of the Sturm–Liouville equation with nonlocal integral conditions at two end-points. We prove that a dense subset of nodal points uniquely determine the potential function of the Sturm–Liouville equation up to a constant.

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In this paper, we give Ambarzumyan-type theorems for a Sturm–Liouville dynamic equation with Robin boundary conditions on a time scale. Under certain conditions, we prove that the potential can be specified from only the first eigenvalue.

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This paper investigates the inverse scattering problem for a discrete Dirac system on the entire line with coefficients that stabilize to zero in one direction. We develop an algorithm for solving the inverse problem of reconstruction of coefficients. We derive a necessary and a sufficient condition on the scattering data so that the inverse problem is uniquely solvable.

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A graph with loops, endowed with standard matching conditions in the internal vertex and with Dirichlet boundary conditions at the boundary vertices, is considered. We show that the potential on a graph with loops can be constructed by the dense nodal points on the interval considered. Moreover, we investigate the so-called incomplete inverse problems of recovering the potential on a fixed edge from a subset of nodal points situated only on a part of the edge.

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Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

Abstract

This paper deals with non-self-adjoint differential operators with two constant delays generated by -y′′+q1(x)y(x-τ1)+(-1)iq2(x)y(x-τ2), where π3τ2<π2<2τ2τ1<π and potentials qj are real-valued functions, qjL2[0,π]. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions y(0)=y(π)=0 and the remaining two under boundary conditions y(0)=y(π)=0.