scholarly journals Recovery of the matrix quadratic differential pencil from the spectral data

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Natalia Bondarenko
Keyword(s):  
Banach Space ◽  
Inverse Problem ◽  
Spectral Data ◽  
Quadratic Differential ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
The Matrix ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Liouville Operators

AbstractWe consider a pencil of matrix Sturm–Liouville operators on a finite interval. We study the properties of its spectral characteristics and inverse problems that consist in the recovering of the pencil by the spectral data, that is, eigenvalues and so-called weight matrices. This inverse problem is reduced to a linear equation in a Banach space by the method of spectral mappings. A constructive algorithm for the solution of the inverse problem is provided.

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 An inverse spectral problem for non-selfadjoint Sturm-Liouville operators with nonseparated boundary conditions

2012 ◽  
Vol 43 (2) ◽  
pp. 289-299 ◽  
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Uniqueness Theorem ◽  
Inverse Spectral Problem ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Nonseparated Boundary Conditions ◽  
Sturm Liouville ◽  
Liouville Operators

Non-selfadjoint Sturm-Liouville operators on a finite interval with nonseparated boundary conditions are studied. We establish properties of the spectral characteristics and investigate an inverse problem of recovering the operators from their spectral data. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing the solution.

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 Spectral analysis of the matrix Sturm–Liouville operator

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Natalia P. Bondarenko
Keyword(s):  
Liouville Operator ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Asymptotic Formulas ◽  
Inverse Spectral Theory ◽  
Different Types ◽  
The Matrix ◽  
Sturm Liouville ◽  
Shaped Graph ◽  
Liouville Operators

AbstractThe self-adjoint matrix Sturm–Liouville operator on a finite interval with a boundary condition in general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on an analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm–Liouville operators on a star-shaped graph with two different types of matching conditions.

An inverse problem for Sturm–Liouville operators on the half-line with complex weights

2019 ◽  
Vol 27 (3) ◽  
pp. 439-443
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Half Line ◽  
Sturm Liouville ◽  
The Given ◽  
Complex Valued ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Abstract Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

An inverse problem for Sturm–Liouville operators with non-separated boundary conditions containing the spectral parameter

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Chinare G. Ibadzadeh ◽  
Ibrahim M. Nabiev
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Spectral Data ◽  
Spectral Parameter ◽  
Boundary Value ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Liouville Operators

AbstractIn this paper a boundary value problem is considered generated by the Sturm–Liouville equation and non-separated boundary conditions, one of which contains a spectral parameter. We give a uniqueness theorem, develop an algorithm for solving the inverse problem of reconstruction of boundary value problems with spectral data. We use the spectra of two boundary value problems and some sequence of signs as a spectral data.

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 A unuqueness theorem for Sturm-Lioville operators with eigenparameter dependent boundary conditions

2012 ◽  
Vol 43 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Yu-Ping Wang
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Uniqueness Theorem ◽  
Linear Function ◽  
Spectral Parameter ◽  
Finite Interval ◽  
Sturm Liouville ◽  
Fractional Linear Function ◽  
Liouville Operators

In this paper, we discuss the inverse problem for Sturm- Liouville operators with boundary conditions having fractional linear function of spectral parameter on the finite interval $[0, 1].$ Using Weyl m-function techniques, we establish a uniqueness theorem. i.e., If q(x) is prescribed on $[0,\frac{1}{2}+\alpha]$ for some $\alpha\in [0,1),$ then the potential $q(x)$ on the interval $[0, 1]$ and fractional linear function $\frac{a_2\lambda+b_2}{c_2\lambda+d_2}$  of the boundary condition are uniquely determined by a subset $S\subset \sigma (L)$ and fractional linear function $\frac{a_1\lambda+b_1}{c_1\lambda+d_1}$ of the boundary condition.

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