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.

scholarly journals An inverse problem for Sturm-Liouville operators on the half-line having Bessel-type singularity in an interior point

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Alexey Fedoseev
Keyword(s):  
Inverse Problem ◽  
Interior Point ◽  
Sufficient Conditions ◽  
Weyl Function ◽  
Half Line ◽  
Type Singularity ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
The Given ◽  
Liouville Operators

AbstractWe study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided, also necessary and sufficient conditions for the solvability of the inverse problem are obtained.

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 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.

scholarly journals Inverse problem for Sturm-Liouville operators on a curve

2019 ◽  
Vol 50 (3) ◽  
pp. 349-359
Author(s):  
Andrey Aleksandrovich Golubkov ◽  
Yulia Vladimirovna Kuryshova
Keyword(s):  
Inverse Problem ◽  
Asymptotic Behavior ◽  
Transfer Matrix ◽  
Liouville Equation ◽  
Inverse Spectral Problem ◽  
Rectifiable Curve ◽  
Uniqueness Of The Solution ◽  
Sturm Liouville ◽  
Discontinuity Conditions ◽  
Liouville Operators

he inverse spectral problem for the Sturm-Liouville equation with a piecewise-entire potential function and the discontinuity conditions for solutions on a rectifiable curve \(\gamma \subset \textbf{C}\) by the transfer matrix along this curve is studied. By the method of a unit transfer matrix the uniqueness of the solution to this problem is proved with the help of studying of the asymptotic behavior of the solutions to the Sturm-Liouville equation for large values of the spectral parameter module.

scholarly journals INVERSE SPECTRAL PROBLEMS FOR STURM–LIOUVILLE OPERATORS WITH SINGULAR POTENTIALS. IV. POTENTIALS IN THE SOBOLEV SPACE SCALE

2006 ◽  
Vol 49 (2) ◽  
pp. 309-329 ◽  
Author(s):  
Rostyslav O. Hryniv ◽  
Yaroslav V. Mykytyuk
Keyword(s):  
Sobolev Space ◽  
Spectral Data ◽  
Sufficient Conditions ◽  
Inverse Spectral Problems ◽  
Singular Potentials ◽  
Spectral Problems ◽  
Sturm Liouville ◽  
Inverse Spectral ◽  
Necessary And Sufficient ◽  
Liouville Operators

AbstractWe solve the inverse spectral problems for the class of Sturm–Liouville operators with singular real-valued potentials from the Sobolev space $W^{s-1}_2(0,1)$, $s\in[0,1]$. The potential is recovered from two spectra or from one spectrum and the norming constants. Necessary and sufficient conditions for the spectral data to correspond to a potential in $W^{s-1}_2(0,1)$ are established.

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.

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