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For any positive integer and any given distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.
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V. A. Yurko 《Proceedings of the Steklov Institute of Mathematics》2006,255(1):260-274
Inverse spectral problems are studied for non-self-adjoint systems of ordinary differential equations on a finite interval.
We establish properties of spectral characteristics and provide a procedure for constructing the solution of the inverse problem
of recovering the coefficients of differential systems from given spectra.
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 273–287. 相似文献
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Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter 下载免费PDF全文
In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of 1 – 3 except for arbitrary one eigenvalue and the Borg type theorem of 1 – 3 except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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V. Yurko 《Journal of Mathematical Analysis and Applications》2006,320(1):439-463
The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem. 相似文献
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We study the inverse spectral problem for a class of Bessel operators given in L2(0,1) by the differential expression
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S.A. Buterin 《Journal of Mathematical Analysis and Applications》2007,335(1):739-749
An inverse spectral problem is studied for a non-selfadjoint Sturm-Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given. 相似文献
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A. A. Abramov V. I. Ul’yanova L. F. Yukhno 《Computational Mathematics and Mathematical Physics》2007,47(5):791-796
A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions. 相似文献
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A. Yu. Chebotarev 《Computational Mathematics and Mathematical Physics》2014,54(3):537-545
An inverse problem for a nonlinear equation in a Hilbert space is considered in which the right-hand side that is a linear combination of given functionals is found from given values of these functionals on the solution. Sufficient conditions for the existence of a solution are established, and the solution set is shown to be homeomorphic to a finite-dimensional compact set. A boundary inverse problem for the three-dimensional thermal convection equations for a viscous incompressible fluid and an inverse magnetohydrodynamics problem are considered as applications. 相似文献
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V. A. Yurko 《Siberian Mathematical Journal》2009,50(2):373-378
We study inverse nodal problems for the second order differential operators on a star-type graph satisfying the standard matching
conditions at the interior vertex. We prove uniqueness theorems and obtain a constructive solution to the inverse problems
of this class.
Original Russian Text Copyright ? 2009 Yurko V. A.
The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00003) and the National Science Council
of Taiwan (Grant 07-01-92000-NSC-a).
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Saratov. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 469–475, March–April, 2009. 相似文献
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Peter C. Gibson 《Transactions of the American Mathematical Society》2002,354(12):4703-4749
We solve the following physically motivated problem: to determine all finite Jacobi matrices and corresponding indices such that the Green's function
is proportional to an arbitrary prescribed function . Our approach is via probability distributions and orthogonal polynomials.
(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
is proportional to an arbitrary prescribed function . Our approach is via probability distributions and orthogonal polynomials.
We introduce what we call the auxiliary polynomial of a solution in order to factor the map
(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
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Chuan-Fu Yang 《Journal of Mathematical Analysis and Applications》2010,365(2):742-554
In this work, we consider inverse spectral problems for the Sturm-Liouville differential operator on a d-star-type graph with standard matching conditions in the internal vertex, where the integer d?2. By using the Yurko's method (Yurko (2008) [27], Yurko (2009) [28]) we show that
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- if the potential function qj(x) on a fixed edge ej is prescribed on the interval , then the reciprocal of d of the spectrum suffices to determine qj(x) on the whole interval [0,π];
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- the 2 over d of the spectrum suffices to determine qj(x) on a fixed edge ej.
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V. A. Yurko 《Differential Equations》2008,44(12):1721-1729
We study boundary value problems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of the spectrum and analyze the inverse problem of reconstructing the coefficients of a differential equation on the basis of the so-called Weyl functions. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mapping. 相似文献