Inverse Sturm-Liouville problem with analytical functions in the boundary condition

Abstract The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

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On the Hochstadt–Lieberman type problem with eigenparameter dependent boundary condition

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2019-0072 ◽

2020 ◽

Vol 28 (4) ◽

pp. 557-565

Author(s):

Sheng-Yu Guan ◽

Chuan-Fu Yang ◽

Natalia Bondarenko ◽

Xiao-Chuan Xu ◽

Yi-Teng Hu

Keyword(s):

Boundary Condition ◽

Inverse Problem ◽

Existence Theorem ◽

Liouville Operator ◽

Entire Functions ◽

Finite Interval ◽

Type Problem ◽

Reconstruction Procedure ◽

Half Inverse Problem ◽

Sturm Liouville

AbstractThe half-inverse problem is studied for the Sturm–Liouville operator with an eigenparameter dependent boundary condition on a finite interval. We develop a reconstruction procedure and prove the existence theorem for solution of the inverse problem. Our method is based on interpolation of entire functions.

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

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.50.2019.2735 ◽

2018 ◽

Vol 50 (1) ◽

pp. 71-102 ◽

Cited By ~ 5

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.

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DUAL EQUATION AND INVERSE PROBLEM FOR AN INDEFINITE STURM–LIOUVILLE PROBLEM WITH M TURNING POINTS OF EVEN ORDER

Mathematical Modelling and Analysis ◽

10.3846/13926292.2012.732972 ◽

2012 ◽

Vol 17 (5) ◽

pp. 618-629

Author(s):

Hamidreza Marasi ◽

Aliasghar Jodayree Akbarfam

Keyword(s):

Inverse Problem ◽

Infinite Product ◽

Turning Points ◽

Liouville Problem ◽

Product Representation ◽

Dual Equation ◽

Number Of Zeros ◽

Even Order ◽

Sturm Liouville ◽

Infinite Product Representation

In this paper the differential equation y″ + (ρ 2 φ 2 (x) –q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and ρ 2 is a real parameter. Also, [0, 1] contains a finite number of zeros of φ 2 , the so called turning points, 0 < x 1 < x 2 < … < x m < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.

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ON POSITIVE EIGENFUNCTIONS OF STURM-LIOUVILLE PROBLEM WITH NONLOCAL TWO‐POINT BOUNDARY CONDITION

Mathematical Modelling and Analysis ◽

10.3846/1392-6292.2007.12.215-226 ◽

2007 ◽

Vol 12 (2) ◽

pp. 215-226 ◽

Cited By ~ 8

Author(s):

Sigita Pečiulytė ◽

Artūras Štikonas

Keyword(s):

Boundary Condition ◽

Nonlocal Boundary ◽

Sufficient Conditions ◽

Liouville Problem ◽

Point Boundary ◽

Existence Domain ◽

Classical Boundary ◽

Sturm Liouville ◽

Necessary And Sufficient ◽

Nonlocal Boundary Problem

Positive eigenvalues and corresponding eigenfunctions of the linear Sturm‐Liouville problem with one classical boundary condition and another nonlocal two‐point boundary condition are considered in this paper. Four cases of nonlocal two‐point boundary conditions are analysed. We get positive eigenfunctions existence domain for each case of these problems. This domain depends on the parameters of the nonlocal boundary problem and it gives necessary and sufficient conditions for existing positive eigenvalues with positive eigenfunctions.

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A solution to the inverse problem for the Sturm-Liouville-type equation with a delay

Filomat ◽

10.2298/fil1307237p ◽

2013 ◽

Vol 27 (7) ◽

pp. 1237-1245 ◽

Cited By ~ 3

Author(s):

Milenko Pikula ◽

Vladimir Vladicic ◽

Olivera Markovic

Keyword(s):

Inverse Problem ◽

Boundary Conditions ◽

Fourier Coefficients ◽

Liouville Problem ◽

Type Equation ◽

Liouville Type ◽

Separated Boundary Conditions ◽

Sturm Liouville ◽

Delay Factor ◽

Spectral Assignment

The paper is devoted to study of the inverse problem of the boundary spectral assignment of the Sturm-Liouville with a delay. -y'(x) + q(x)y(? ? x) = ?y(x), q ? AS[0, ?], ? ? (0,1] (1) with separated boundary conditions: y(0) = y(?) = 0 (2) y(0) = y'(?) = 0 (3) It is argued that if the sequence of eigenvalues is given ?n(1) and ?n(2) tasks (1-2) and (1-3) respectively, then the delay factor ? ? (0,1) and the potential q ? AS[0, ?] are unambiguous. The potential q is composed by means of trigonometric Fourier coefficients. The method can be easily transferred to the case of ? = 1 i.e. to the classical Sturm-Liouville problem.

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

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.50.2019.3347 ◽

2019 ◽

Vol 50 (3) ◽

pp. 207-221 ◽

Cited By ~ 2

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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Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

Mathematics ◽

10.3390/math9202617 ◽

2021 ◽

Vol 9 (20) ◽

pp. 2617

Author(s):

Natalia P. Bondarenko ◽

Andrey V. Gaidel

Keyword(s):

Inverse Problem ◽

Inverse Spectral Problem ◽

Sufficient Conditions ◽

Local Solvability ◽

Global Solvability ◽

Multiple Eigenvalues ◽

Eigenvalue Multiplicities ◽

Complex Valued ◽

Theoretical Results ◽

Infinite Sequences

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

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