scholarly journals Solving An Inverse Problem For The Sturm-Liouville Operator With A Singular Potential By Yurko's Method

2020 ◽  
Vol 52 (1) ◽  
Author(s):  
Natalia P. Bondarenko
Keyword(s):  
Inverse Problem ◽  
Liouville Operator ◽  
Singular Potential ◽  
Inverse Spectral Problem ◽  
Sufficient Conditions ◽  
The Self ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
The Uniqueness Theorem

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases.

scholarly journals An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

2018 ◽  
Vol 50 (1) ◽  
pp. 71-102 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problem ◽  
Liouville Operator ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Sufficient Conditions ◽  
Adjoint Matrix ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient

The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is given as a corollary of the main result.

On recovering a Sturm–Liouville-type operator with the frozen argument rationally proportioned to the interval length

2019 ◽  
Vol 27 (3) ◽  
pp. 429-438 ◽  
Author(s):  
Sergey A. Buterin ◽  
Sergey V. Vasiliev
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Interval Length ◽  
Liouville Type ◽  
Type Property ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Complex Valued ◽  
The Uniqueness Theorem

Abstract We consider the operator {\ell y\mathrel{\mathop{:}}=-y^{\prime\prime}(x)+q(x)y(a)} , {0<x<\pi} , {y(0)=y(\pi)=0} , where {q(x)\in L_{2}(0,\pi)} is a complex-valued function and {a/\pi\in[0,1]} is a rational number. The inverse problem of recovering the potential {q(x)} from the spectrum of {\ell} is studied. We describe the sets of iso-spectral potentials and prove the uniqueness theorem in the class of potentials possessing some symmetry-type property. Moreover, we obtain a constructive procedure for solving this inverse problem along with necessary and sufficient conditions of its solvability, which in turn give the characterization of the spectrum. In parallel, we establish that the informativity of the spectrum is severely unstable with respect to the parameter a.

scholarly journals Necessary and sufficient conditions for the solvability of the inverse problem for non-self-adjoint pencils of Sturm-Liouville operators on the half-line

2011 ◽  
Vol 42 (3) ◽  
pp. 247-258 ◽  
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Weyl Function ◽  
Half Line ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Liouville Operators

Non-self-adjoint Sturm-Liouville differential operators on the half-line with a boundary condition depending polynomially on the spectral parameter are studied. We investigate the inverse problem of recovering the operator from the Weyl function. For this inverse problem we provide necessary and suffcient conditions for its solvability along with a procedure for constructing its solution by the method of spectral mappings.

scholarly journals Spectral analysis for the matrix Sturm-Liouville operator on a finite interval

2011 ◽  
Vol 42 (3) ◽  
pp. 305-327 ◽  
Author(s):  
Natalia Bondarenko
Keyword(s):  
Inverse Problem ◽  
Spectral Analysis ◽  
Liouville Equation ◽  
Inverse Spectral Problem ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Sufficient Conditions ◽  
The Matrix ◽  
Sturm Liouville ◽  
Necessary And Sufficient

The inverse spectral problem is investigated for the matrix Sturm-Liouville equation on a finite interval. Properties of spectral characteristics are provided, a constructive procedure for the solution of the inverse problem along with necessary and sufficient conditions for its solvability is obtained.

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 Inverse Spectral Problem for Integrodifferential Sturm-Liouville Operators with Discontinuity Conditions

2018 ◽  
Vol 64 (3) ◽  
pp. 427-458 ◽  
Author(s):  
S A Buterin
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Global Solvability ◽  
Convolution Integral ◽  
Main Equation ◽  
Convolution Integral Operator ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Discontinuity Conditions ◽  
Liouville Operators

We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodifferential expression under consideration. We prove the global solvability of the main equation, this implies the uniqueness of solution of the inverse problem and leads to necessary and sufficient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem.

Sign in / Sign up

Export Citation Format

Share Document