Matrix Sturm-Liouville Operator Research Articles

Inverse spectral problems for functional-differential operators with involution

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems.

Inverse problem solution and spectral data characterization for the matrix Sturm–Liouville operator with singular potential

The self-adjoint matrix Sturm–Liouville operator on a finite interval with singular potential of class $$W_2^{-1}$$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm–Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for the studied operator. In addition, the main results are applied to the Sturm–Liouville operator on a graph of arbitrary geometrical structure.

Direct and Inverse Problems for the Matrix Sturm–Liouville Operator with General Self-Adjoint Boundary Conditions

The matrix Sturm–Liouville operator on a finite interval with boundary conditions in general self-adjoint form and with singular potential of class $$W_2^{-1}$$ is studied. This operator generalizes Sturm–Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.

Spectral data characterization for the Sturm–Liouville operator on the star-shaped graph

The inverse spectral problems are studied for the Sturm–Liouville operator on the star-shaped graph and for the matrix Sturm–Liouville operator with one 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.

Space versus energy oscillations of Prufer phases for matrix Sturm-Liouville and Jacobi operators

This note considers Sturm oscillation theory for regular matrix Sturm-Liouville operators on finite intervals and for matrix Jacobi operators. The number of space oscillations of the eigenvalues of the matrix Prufer phases at a given energy, defined by a suitable lift in the Jacobi case, is shown to be equal to the number of eigenvalues below that energy. This results from a positivity property of the Prufer phases, namely they cannot cross -1 in the negative direction, and is also shown to be closely linked to the positivity of the matrix Prufer phase in the energy variable. The theory is illustrated by numerical calculations for an explicit example.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/76/abstr.html

Constructive solution of the inverse spectral problem for the matrix Sturm–Liouville operator

An inverse spectral problem is studied for the matrix Sturm–Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the method of spectral mappings for the considered inverse problem. The nonlinear inverse problem is reduced to a linear equation in a special Banach space of infinite matrix sequences. In addition, we apply our results to the Sturm–Liouville operator on a star-shaped graph.

Spectral analysis of the matrix Sturm\u2013Liouville operator

The 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.

Inverse spectral problem for the matrix Sturm-Liouville operator with the general separated self-adjoint boundary conditions

In this work, we study the matrix Sturm-Liouville operator with the separated self-adjoint boundary conditions of general type, in terms of two unitary matrices. Some properties of the eigenvalues and the normalization matrices are given. Uniqueness theorems for determining the potential and the unitary matrices in the boundary conditions from the Weyl matrix, two characteristic matrices or one spectrum and the corresponding normalization matrices are proved.

Sampling eigenvalues by Hermite revisited

ABSTRACTWe show how the Hermite sampling method can be used to approximate the eigenvalues of Sturm-Liouville problems. Because it involves the derivatives with respect to the eigenvalue parameter, it requires more smoothness on the coefficients and integrating a one parameter family of matrix Sturm-Liouville operators to generate the required samples.

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

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.

Limit-point criteria for the matrix Sturm-Liouville operator and its powers

We consider matrix Sturm-Liouville operators generated by the formal expression \[l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,\] in the space \(L^2_n(I)\), \(I:=[0, \infty)\). Let the matrix functions \(P:=P(x)\), \(Q:=Q(x)\) and \(R:=R(x)\) of order \(n\) (\(n \in \mathbb{N}\)) be defined on \(I\), \(P\) is a nondegenerate matrix, \(P\) and \(Q\) are Hermitian matrices for \(x \in I\) and the entries of the matrix functions \(P^{-1}\), \(Q\) and \(R\) are measurable on \(I\) and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices \(P\), \(Q\) and \(R\) that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by \(l^k[y]\) (\(k \in \mathbb{N}\)). In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.

Trace formula for the matrix Sturm-Liouville operator

A regularized trace formula of first order for the matrix Sturm-Liouville operator \(-\frac{d^2}{dx^2}+Q\) with a \(d\times d\) self-adjoint matrix-valued potential \(Q\) in \(L^2((0,\pi );dx)^d\) is derived, which extends the well-known scalar case \(d=1\).

Recovery of the matrix quadratic differential pencil from the spectral data

Abstract We 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.

An inverse spectral problem for the matrix Sturm-Liouville operator on the half-line

The matrix Sturm-Liouville operator with an integrable potential on the half-line is considered. The inverse spectral problem is studied, which consists in recovering of this operator by the Weyl matrix. The author provides necessary and sufficient conditions for a meromorphic matrix function being a Weyl matrix of the non-self-adjoint matrix Sturm-Liouville operator. We also investigate the self-adjoint case and obtain the characterization of the spectral data as a corollary of our general result.

Non-Selfadjoint Matrix Sturm-Liouville Operators with Eigenvalue-Dependent Boundary Conditions

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L (R+, S) by the differential expression ` (y) = −y +Q (x) y , x ∈ R+ : [0,∞) , and the boundary condition y′ (0)− ( β0 + β1λ+ β2λ 2 ) y (0) = 0 where Q is a non-selfadjoint matrix valued function. Also using the uniqueness theorem of analytic functions we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities. 2000 AMS Classification: 34B24, 34L40 , 47A10.

Erratum to “Necessary and Sufficient Conditions for the Solvability of the Inverse Problem for the Matrix Sturm-Liouville Operator”

The matrix Sturm-Liouville operator on a finite interval with Dirichlet boundary conditions is studied. Properties of its spectral characteristics and the inverse problem of recovering the operator from these characteristics are investigated. Necessary and sufficient conditions on the spectral data of the operator are obtained. Research is conducted in the general case, with no a priori restrictions on the spectrum. A constructive algorithm for solving the inverse problem is provided.

Integration of the matrix KdV equation with self-consistent source

In this work, it is shown that the solutions of the matrix KdV equation with self-consistent source can be found by the inverse scattering transform for the matrix Sturm–Liouville operator.

An inverse problem for the quadratic pencil of non-self-adjoint matrix operators on the half-line

Abstract We consider a pencil of non-self-adjoint matrix Sturm–Liouville operators on the half line and study the inverse problem of constructing this pencil by its Weyl matrix. A uniqueness theorem is proved, and a constructive algorithm for the solution is obtained.

On Ambarzumyan-type theorems

Generalizations of the classical Ambarzumyan theorem are provided for wide classes of self-adjoint differential operators with arbitrary self-adjoint boundary conditions: scalar Sturm–Liouville operators, higher-order differential operators, matrix Sturm–Liouville operators and operators on spatial networks.

Necessary and sufficient conditions for the solvability of the inverse problem for the matrix Sturm-Liouville operator

Necessary and sufficient conditions for the solvability of the inverse problem for the matrix Sturm-Liouville operator