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1.
In this paper we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of the considered problem coincide with those of A. We obtain asymptotic formulas for the eigenvalues and eigenfunctions. Also we show that the eigenelements of A are complete in H.  相似文献   

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We study the finite spectrum of Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions. For any positive integers m and n, we construct a class of regular Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, which have at most m + n + 4 eigenvalues.  相似文献   

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We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.  相似文献   

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This paper is devoted to the calculation of the deficiency index of a differential operator. In particular, we present sufficient conditions under which the operator with homogeneous boundary condition at zero is self-adjoint.  相似文献   

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We prove that a differential boundary operator of the Sturm–Liouville type on a semiaxis with two-point integral boundary conditions that acts in the Hilbert space L 2(0, ) is closed and densely defined. The adjoint operator is constructed. We also establish criteria for the maximal dissipativity and maximal accretivity of this operator.  相似文献   

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Bondarenko  N. P. 《Mathematical Notes》2021,109(3-4):358-378
Mathematical Notes - The matrix Sturm–Liouville operator on a finite interval with boundary conditions in general self-adjoint form and with singular potential of class $$W_2^{-1}$$ is...  相似文献   

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The purpose of this paper is to study a Sturm–Liouville problem with discontinuities in the case when an eigenparameter appears not only in the differential equation but it also appears in both the boundary and transmission conditions. We suggest a new approach for the definition of a suitable Hilbert space and a symmetric linear operator defined in this space in such a way that the considered problem can be interpreted as the eigenvalue problem of this operator and for construction and approximation of a fundamental solution. We apply these results to find asymptotic formulas of eigenvalues and corresponding eigenfunctions. Mathematics Subject Classification (2000) 34L20.This work was supported by the Research Fund of Gaziosmanpasa University under grand no:2004/01.  相似文献   

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In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data.  相似文献   

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This paper is concerned with the application of the Kramer sampling theorem to Sturm–Liouville problems with coupled boundary conditions. The analysis is restricted to the case when the spectrum of the boundary value problem is simple. In all such cases, it is shown that Kramer analytic kernels can be defined and that each kernel has an associated analytic interpolation function to give the Lagrange interpolation series.  相似文献   

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The spectral problem for the Sturm–Liouville operator with an arbitrary complex-valued potential q(x) of the class L1(0, π) and degenerate boundary conditions is considered. We prove that the system of root functions of this operator is not a basis in the space L2(0, π).  相似文献   

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Osilenker  B. P. 《Mathematical Notes》2020,108(5-6):842-853
Mathematical Notes - Potential functions associated with eigenfunctions of the discrete Sturm–Liouville operator are studied in a loaded space. On the basis of representations of kernels with...  相似文献   

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Doklady Mathematics - We study the equiconvergence of spectral decompositions for two Sturm–Liouville operators on the interval [0, π] generated by the differential expressions...  相似文献   

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For semicontinuous summation methods generated by Λ = {λn(h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class B, some results on the uniform a.e. behavior of Λ-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means λn(h) = exp(?uα(n)h) are obtained. An application to a generalized heat equation is given.  相似文献   

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Dissipative singular Sturm–Liouville operators are studied in the Hilbert space Lw2[a,b) (–<a<b), that the extensions of a minimal symmetric operator in Weyls limit-point case. We construct a selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of a selfadjoint operator. Finally, in the case when the Titchmarsh–Weyl function of the selfadjoint operator is a meromorphic in complex plane, we prove theorems on completeness of the system of eigenfunctions and associated functions of the dissipative Sturm–Liouville operators. Mathematics Subject Classifications (2000) 47A20, 47A40, 47A45, 34B20, 34B44, 34L10.  相似文献   

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