93 research outputs found
Eigenvalue asymptotics for Sturm--Liouville operators with singular potentials
We derive eigenvalue asymptotics for Sturm--Liouville operators with singular
complex-valued potentials from the space W^{\al-1}_{2}(0,1), \al\in[0,1],
and Dirichlet or Neumann--Dirichlet boundary conditions. We also give
application of the obtained results to the inverse spectral problem of
recovering the potential from these two spectra.Comment: Final version as appeared in JF
On impulsive Sturm–Liouville operators with singularity and spectral parameter in boundary conditions
We study properties and the asymptotic behavior of spectral characteristics for a class of singular Sturm–Liouville differential operators with discontinuity conditions and an eigenparameter in boundary conditions. We also determine theWeyl function for this problem and prove uniqueness theorems for a solution of the inverse problem corresponding to this function and spectral data.Дослiджено властивостi та асимптотичну поведiнку спектральних характеристик для класу сингулярних диференцiальних операторiв Штурма – Лiувiлля з розривними умовами та власним параметром у граничних умовах. Визначено функцiю Вейля для цiєї задачi та доведено теореми про єдинiсть розв’язку оберненої задачi, що вiдповiдає цiй функцiї та спектральним даним
On the one dimensional Gelfand and Borg-Levinson spectral problems for discontinuous coefficients
In this paper, we deal with the inverse spectral problem for the equation -(pu')'+qu = \lambda\rho u on a finite interval (0; h). We give some uniqueness results on q and \rho from the Gelfand spectral data, when the coefficients p and \rho are piecewise Lipschitz and q is bounded. We also prove an equivalence result between the Gelfand spectral data and the Borg-Levinson spectral data. As a consequence, we have similar uniqueness results if we consider the Borg-Levinson spectral data. Finally, we consider the inverse problem from the nodes and give uniqueness results on \rho and in the case where the coefficients p; q and \rho are smooth we give a uniqueness results on both q and \rho
Forward and inverse spectral theory of Sturm-Liouville operators with transmission conditions
Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, School of Mathematics, 2017.ForwardandinversespectralproblemsconcerningSturm-Liouvilleoperatorswithoutdiscontinuitieshavebeenstudiedextensively. Bycomparison,therehasbeenlimitedworktacklingthecase where the eigenfunctions have discontinuities at interior points, a case which appears naturally in physical applications. We refer to such discontinuity conditions as transmission conditions. We consider Sturm-Liouville problems with transmission conditions rationally dependent on the spectral parameter. We show that our problem admits geometrically double eigenvalues, necessitating a new analysis. We develop the forward theory associated with this problem and also consider a related inverse problem. In particular, we prove a uniqueness result analogous to that of H. Hochstadt on the determination of the potential from two sequences of eigenvalues. In addition, we consider the problem of extending Sturm’s oscillation theorem, regarding the number of zeroes of eigenfunctions, from the classical setting to discontinuous problems with general constant coefficient transmission conditionsGR201
Sturm-Liouville operators with measure-valued coefficients
We give a comprehensive treatment of Sturm-Liouville operators with
measure-valued coefficients including, a full discussion of self-adjoint
extensions and boundary conditions, resolvents, and Weyl-Titchmarsh theory. We
avoid previous technical restrictions and, at the same time, extend all results
to a larger class of operators. Our operators include classical Sturm-Liouville
operators, Lax operators arising in the treatment of the Camassa-Holm equation,
Jacobi operators, and Sturm-Liouville operators on time scales as special
cases.Comment: 58 page
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