Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues

2020 ◽  
Vol 28 (3) ◽  
pp. 341-348 ◽  
Author(s):  
Ran Zhang ◽  
Xiao-Chuan Xu ◽  
Chuan-Fu Yang ◽  
Natalia Pavlovna Bondarenko
Keyword(s):  
Boundary Conditions ◽  
Interior Point ◽  
Liouville Operator ◽  
Inverse Spectral Problem ◽  
Liouville Problem ◽  
Robin Boundary Conditions ◽  
Jump Conditions ◽  
Sturm Liouville ◽  
Robin Boundary

AbstractIn this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on {(0,\pi)} with the Robin boundary conditions and the jump conditions at the point {\frac{\pi}{2}}. We prove that the potential {M(x)} on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential {M(x)} is given on {(0,\frac{(1+\alpha)\pi}{4})}; (ii) the potential {M(x)} is given on {(\frac{(1+\alpha)\pi}{4},\pi)}, where {0<\alpha<1}, respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point.

Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem

2019 ◽  
Vol 22 (1) ◽  
pp. 78-94 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Continuous Functions ◽  
Real Spectrum ◽  
Robin Boundary Conditions ◽  
Schmidt Operator ◽  
Sturm Liouville ◽  
Robin Boundary

Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

scholarly journals An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 207-221 ◽  
Author(s):  
Sergey Buterin
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
A Priori ◽  
Sufficient Conditions ◽  
Robin Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

scholarly journals An Inverse Spectral Problem for the Sturm-Liouville Operator on a Three-Star Graph

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
I. Dehghani Tazehkand ◽  
A. Jodayree Akbarfam
Keyword(s):  
Inverse Problem ◽  
Spectral Problem ◽  
Liouville Operator ◽  
Inverse Spectral Problem ◽  
Spectral Characteristics ◽  
Star Graph ◽  
Dirichlet Problems ◽  
Sturm Liouville ◽  
Inverse Spectral ◽  
Robin Boundary

We study an inverse spectral problem for the Sturm-Liouville operator on a three-star graph with the Dirichlet and Robin boundary conditions in the boundary vertices and matching conditions in the internal vertex. As spectral characteristics,we consider the spectrum of the main problem together with the spectra of two Dirichlet-Dirichlet problems and one Robin-Dirichlet problem on the edges of the graph and investigate their properties and asymptotic behavior. We prove that if these four spectra do not intersect, then the inverse problem of recovering the operator is uniquely solvable.We give an algorithm for the solution of the inverse problem with respect to this quadruple of spectra.

scholarly journals Ambarzumyan-type theorems on a time scale

2018 ◽  
Vol 26 (5) ◽  
pp. 633-637 ◽  
Author(s):  
Ahmet Sinan Ozkan
Keyword(s):  
Boundary Conditions ◽  
Dynamic Equation ◽  
Time Scale ◽  
Robin Boundary Conditions ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Sturm Liouville ◽  
Robin Boundary

Abstract 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.

On the determination of the impulsive Sturm–Liouville operator with the eigenparameter‐dependent boundary conditions

2020 ◽  
Vol 43 (12) ◽  
pp. 7143-7151 ◽  
Author(s):  
Yasser Khalili ◽  
Nematollah Kadkhoda ◽  
Dumitru Baleanu
Keyword(s):  
Boundary Conditions ◽  
Liouville Operator ◽  
Sturm Liouville

scholarly journals Solution of a model inverse spectral problem for the Sturm-Liouville operator on a graph

2016 ◽  
pp. 204-211
Author(s):  
Н.Ф. Валеев ◽  
Ю.В. Мартынова ◽  
Я.Т. Султанаев
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Liouville Operator ◽  
Inverse Spectral Problem ◽  
Dimensional Space ◽  
Geometric Graph ◽  
Finite Dimensional Space ◽  
Sturm Liouville ◽  
Inverse Spectral ◽  
Model Inverse

Исследуется модельная обратная спектральная задача для оператора Штурма-Лиувилля на геометрическом графе. Суть данной задачи состоит в восстановлении $N$ параметров граничных условий по $N$ собственным значениям. Установлено, что эта задача обладает свойством монотонной зависимости собственных значений от параметров граничных условий. Поставленная задача сведена к многопараметрической обратной спектральной задаче для оператора в конечномерном пространстве. Предложен новый алгоритм численного решения рассматриваемой задачи. A model inverse spectral problem for the Sturm-Liouville operator on a geometric graph is studied. This problem consists in finding $N$ parameters of the boundary conditions using its $N$ known eigenvalues. It is shown that the problem under consideration possess the property of a monotonic dependence of its eigenvalues on the parameters of the boundary conditions. This problem is reduced to a multiparameter inverse spectral problem for the operator in a finite-dimensional space. A new algorithm for the numerical solution of this problem is proposed.

scholarly journals Inverse spectral problem for the matrix Sturm-Liouville operator with the general separated self-adjoint boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 321-336 ◽  
Author(s):  
Xiao-Chuan Xu
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Liouville Operator ◽  
General Type ◽  
Inverse Spectral Problem ◽  
Uniqueness Theorems ◽  
Unitary Matrices ◽  
Weyl Matrix ◽  
The Matrix ◽  
Sturm Liouville

In this work, we study the matrix Sturm-Liouville operator with the separated self-adjoint boundary conditions of general type, in terms of two unitary matrices. Some properties of the eigenvalues and the normalization matrices are given. Uniqueness theorems for determining the potential and the unitary matrices in the boundary conditions from the Weyl matrix, two characteristic matrices or one spectrum and the corresponding normalization matrices are proved.

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