Half-inverse Problem Research Articles

Half inverse problem and interior inverse problem for the Dirac operators with discontinuity

Half inverse problem and interior inverse problem for the Dirac operators with discontinuity

Local stability of half inverse problems with boundary conditions dependent on the spectral parameter

In the present paper, the half inverse problem for Sturm-Liouville operator with boundary conditions dependent on the eigenparameter is investigated. It is shown that the potential function can be found on the interval (0,π) or on its subinterval from given the eigenvalues of the problem, and the local stability of the half inverse problem is proved.

Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient

Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

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.

Hochstadt-Lieberman type theorems for a diffusion operator on a time scale

In this paper, we consider a half inverse problem for a diffusion operator on a time scale which is the union of an interval and another arbitrary time scale such as . We give a Hochstadt-Lieberman type theorem for this problem and some appropriate examples.

Reconstruction and solvability for discontinuous Hochstadt–Lieberman problems

We consider Sturm–Liouville problems with a discontinuity in an interior point, which are motivated by the inverse problems for the torsional modes of the Earth. We assume that the potential on the right half-interval and the coefficient in the right boundary condition are given. Half-inverse problems are studied, that consist in recovering the potential on the left half-interval and the left boundary condition from the eigenvalues. If the discontinuity belongs to the left half-interval, the position and the parameters of the discontinuity also can be reconstructed. In this paper, we provide reconstructing algorithms and prove existence of solutions for the considered inverse problems. Our approach is based on interpolation of entire functions.

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

Abstract The 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.

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

AbstractThe 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.

Inverse problems for a conformable fractional Sturm–Liouville operator

Abstract In this paper, a Sturm–Liouville boundary value problem which includes conformable fractional derivatives of order α, 0 < α ≤ 1 {0<\alpha\leq 1} is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study the half-inverse problem and prove a Hochstadt–Lieberman-type theorem.

A half-inverse problem for the singular diffusion operator with jump conditions

In this paper, half inverse spectral problem for diffusion operator with jump conditions dependent on the spectral parameter and discontinuoty coeffcient is considered. The half inverse problems is studied of determining the coeffcient and two potential functions of the boundary value problem its spectrum by Hocstadt- Lieberman and Yang-Zettl methods. We show that two potential functions on the whole interval and the parameters in the boundary and jump conditions can be determined from spectrum.

Half-Inverse Problem For Dirac Operator With Boundary And Transmission Conditions Dependent Spectral Parameter Polynomially

In this paper, half-inverse problem is considered for Dirac equations with boundary and finite number of transmission conditions depending polynomially on the spectral parameter, if the potential is given over the half of the considered interval and if one spectrum is known then, potential function on the whole interval and the other coefficients of the considered problem can be determined uniquely.

Half-inverse problem for the Dirac operator

Given the spectrum of the Dirac operator, together with the potential on the half-interval and one boundary condition, this paper provides reconstruction of the potential on the whole interval, and proves the existence conditions of the solution.

On the half inverse spectral problem for an integro-differential operator

We consider the Sturm–Liouville differential operator on a finite interval with Dirichlet boundary conditions perturbed with a convolution integral operator. The inverse problem is studied of recovering the convolution kernel from a half spectrum, provided that the kernel function on the first half of the interval as well as the Sturm–Liouville potential on the entire interval are known a priori. Besides the uniqueness of a solution of this half inverse problem, we also obtain necessary and sufficient conditions of its solvability. A constructive procedure for solving the inverse problem is also provided.

Half inverse problem for the impulsive diffusion operator with discontinuous coefficient

The half inverse problem is to construct coefficients of the operator in a whole interval by using one spectrum and potential known in a semi interval. In this paper, by using the Hocstadt-Lieberman and Yang-Zettl?s methods we show that if p(x) and q(x) are known on the interval (?/2,?), then only one spectrum suffices to determine p (x),q(x) functions and ?,h coefficients on the interval (0,?) for impulsive diffusion operator with discontinuous coefficient.

On the Hochstadt-Lieberman theorem for discontinuous boundary-valued problems

In this paper, we discuss the half inverse problem for Sturm-Liouville equations with boundary conditions dependent on the spectral parameter and a finite number of discontinuities inside the interval and prove the Hochstadt-Liberman type theorem for the above boundary-valued problem.

Half Inverse Problem for the Sturm-Liouville Operator with Coulomb potential

In this paper, we discuss a half inverse problem for the Sturm-Liouville operator with Coulomb potential and show that if q(x) is prescribed on ( π ,π) then only one spectrum is sufficient to determine potential function q(x) on the whole interval (0,π).

A HOCHSTADT-LIEBERMAN THEOREM FOR INTEGRO-DIFFERENTIAL OPERATOR

In this study, we solve a half inverse problem for integro-differential operator that consists of Sturm-Liouville differential part and integral part of Volterra type on a finite interval by using Hochstadt-Lieberman's method. We consider that q is a squared integrable function on (0,�) and the kernel of the integral perturbation is integrable on its domain. Especially, we show that the potential function q is known as uniquely when the kernel of the integral perturbation is given.

Half-inverse Sturm-Liouville problem with boundary and discontinuity conditions dependent on the spectral parameter

In this article, an impulsive Sturm–Liouville boundary value problem with the eigenvalue parameter rationally contained in one-boundary condition and linearly contained in the jump conditions is considered. It is proven that, if is prescribed on , a single spectrum, excluding a finite number of eigenvalues, is sufficient to determine on the interval and the other coefficients of the problem.

UNIQUENESS OF THE SOLUTION OF HALF INVERSE PROBLEM FOR THE IMPULSIVE STURM LIOUVILLE OPERATOR

Abstract. The half-inverse spectral problem for an impulsive Sturm–Liouville operator consists in reconstruction of this operator from itsspectrum and half of the potential. In this study, the spectrum of the im-pulsive Sturm–Liouville problem is given and by using the Hochstadt andLieberman’s method we show that if q(x) is prescribed on0, π2 , thenonly one spectrum is sufficient to determine q(x) on the interval (0,π) forthis problem 1. IntroductionAssume that Q(x) is a real-valued function in L 2 (0,π),αand βare realnumbers such that 0 0. Define ρ(x) =ˆ1, x π2 anddenote by L= L(Q(x),ρ(x),β) a Sturm–Liouville problem in L 2 (0,π) that isgiven by the differential equation(1) ly:= −y ′′ (x) +Q(x)y(x) = λρ(x)y(x), x∈ (0,π2)∪ (π2,π)with the boundary conditions(2) U(y) := y ′ (0) = 0,(3) V(y) := y ′ (π) = 0,and the jump conditions(4)y(π2+0) = βy(π2−0) ,y ′ (π2+0) = β −1 y ′ (π2−0),where λis the spectral parameter. Received October 25, 2011; Revised February 22, 2012.2010 Mathematics Subject Classification. 34A55, 34B24, 34L05, 45C05.Key words and phrases. Sturm-Liouville operator, determination of the potential, discon-tinuous condition, half inverse problem.

Half inverse problem for Sturm-Liouville operators with boundary conditions dependent on the spectral parameter

In this paper, we discuss the half inverse problem for the Sturm-Liouville operator with boundary conditions dependent on the spectral parameter and show that if q(x) is prescribed on [\frac{\pi}{2},\pi], then one spectrum is sufficient to determine the potential q(x) on the whole interval [0,\pi] and coefficient function \frac{a_1\lambda+b_1}{c_1\lambda+d_1} of the boundary condition.