Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm for solving the inverse problems. The key role in our method is played by the Riesz-basis property of a special vector-functional system in a Hilbert space. In addition, we obtain a new uniqueness theorem for recovering the potential on a part of the interval, by using a fractional part of the spectrum.
In this work, we use the \textit{regularized sampling method} to compute the eigenvalues of Sturm Liouville problems with discontinuity conditions inside a finite interval. We work out an example by computing a few eigenvalues and their corresponding eigenfunctions.
We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local solvability and stability of this inverse problem, relying on the method of spectral mappings. Possible splitting of multiple eigenvalues is taken into account.
We consider the inverse Sturm-Liouville problem with one discontinuous point on three-star graph, we deduced the distribution of the eigenvalues, and proved that one spectrum could uniquely determine the unknown potential and jump information when we know the whole potentials on two edges and half potential on another edge.
The inverse spectral problems are studied for the Sturm-Liouville operator on the star-shaped graph and for the matrix Sturm-Liouville operator with the boundary condition in the general self-adjoint form. We obtain necessary and sufficient conditions of solvability for these two inverse problems, and also prove their local solvability and stability.
In this work, we consider the Sturm-Liouville operator on a finite interval $[0,1]$ with discontinuous conditions at $1/2$. We prove that if the potential is known a priori on a subinterval $[b,1]$ with $b\ge1/2$, then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions. For the case $b<1/2$, parts of either one or two spectra can uniquely determine the potential and a part of parameters.
In this study, the theorem on necessary and sufficient conditions for the solvability of inverse problem for Sturm-Liouville operator with discontinuous coefficient is proved and the algorithm of reconstruction of potential from spectral data (eigenvalues and normalizing numbers) is given.
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.
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 r…
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
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.
This paper is concerned with continuous dependence of the n-th eigenvalue on self-adjoint discrete Sturm-Liouville problems. The n-th eigenvalue is considered as a function in the space of the problems. A necessary and sufficient condition for all the eigenvalue functions to be continuous and several properties of the eigenvalue functions in a set of the space of the problems are given. They play an important role in the study of continuous dependence of the n-th eigenvalue function on the problems. Continuous dependence of the n-th eigenvalue function on the equations and on the boundary cond…
We consider second order linear differential operators possessing a term depending on the unknown function with a fixed argument and study the uniqueness of recovering the operators from the spectrum. We also obtain a constructive procedure for solving this inverse problem along with necessary and sufficient conditions of its solvability.
The Sturm-Liouville operator on a star-shaped graph is considered. We assume that the potential is known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential on the remaining edge from the part of the spectrum. A constructive method is developed for the solution of this problem, based on the Riesz-basicity of some sequence of vector functions. The local solvability of the inverse problem and the stability of its solution are proved.
The perturbation of the Sturm-Liouville operator on a finite interval with Dirichlet boundary conditions by a convolution operator is considered. Local stability and global unique solvability of the inverse problem of recovering the convolution kernel from the spectrum, provided that the potential is given a priori, is known. In the present work, we establish uniform full stability of this inverse problem involving a uniform estimate of deviations of the convolution kernel via deviations of the spectrum and the potential within balls of any fixed radii.
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 derived as a corollary of the main result.
In this paper, inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems are established. For a fixed discrete Sturm-Liouville equation, inequalities among eigenvalues for different boundary conditions are given. For a fixed boundary condition, inequalities among eigenvalues for different equations are given. These results are obtained by applying continuity and discontinuity of the n-th eigenvalue function, monotonicity in some direction of the n-th eigenvalue function, which were given in our previous papers, and natural loops in the space of boundary conditi…
Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.
Half-inverse spectral problem for a Sturm--Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. We give the necessary and sufficient conditions for solvability of the half-inverse spectral problem for the class of Sturm--Liouville operators with singular potentials from the space $W^{-1}_2(0,1)$ and provide the reconstruction algorithm.
We 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.
We consider the boundary problem -y''(x)+q(x)y(x)=f(x), lim_{|x|\to\infty}y^{(i)}(x)=0, i=0,1, where f(x)\in L_p(R), p\in[1,\infty], 1\le q(x)\in L_1^{\loc}(R). For this boundary problem we obtain: 1) necessary and sufficient conditions for unique solvability and a priori properties of the solution; 2) a criterion for the resolvent to be compact in L_p(R), and some a priori properties of the spectrum.
This work deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs.
Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.
We consider a second order functional-differential pencil with two constant delays of the argument and study the inverse problem of recovering its coefficients from the spectra of two boundary value problems with one common boundary condition. The uniqueness theorem is proved and a constructive procedure for solving this inverse problem along with necessary and sufficient conditions for its solvability is obtained. Moreover, we give a survey on the contemporary state of the inverse spectral theory for operators with delay. The pencil under consideration generalizes Sturm-Liouville-type operato…
