Matrix Sturm Liouville Research Articles

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.

Non-compact Quantum Graphs with Summable Matrix Potentials

Let $\mathcal{G}$ be a metric noncompact connected graph with finitely many edges. The main object of the paper is the Hamiltonian ${\bf H}_{\alpha}$ associated in $L^2(\mathcal{G};\mathbb{C}^m)$ with a matrix Sturm-Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian ${\bf H}_{\alpha}$ as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of ${\bf H}_{\alpha}$. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of ${\bf H}_{\alpha}$ is obtained. Additionally, for a star graph $\mathcal{G}$ a formula is found for the scattering matrix of the pair $\{{\bf H}_{\alpha}, {\bf H}_D\}$, where ${\bf H}_D$ is the Dirichlet operator on $\mathcal{G}$.

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.

Quantum Graphs with Summable Matrix Potentials

Let $$\mathcal{G}$$ be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume that the length of at least one of the edges is infinite. The main object of this paper is the Hamiltonian $${{{\mathbf{H}}}_{\alpha }}$$ associated in $${{L}^{2}}(\mathcal{G};{{\mathbb{C}}^{m}})$$ with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian $${{{\mathbf{H}}}_{\alpha }}$$ and any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring that the positive part of $${\mathbf{H}_{\alpha }}$$ is purely absolutely continuous. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of the operator $${{{\mathbf{H}}}_{\alpha }}$$ is obtained. Additionally, a formula is found for the scattering matrix of the pair $$\{ {{{\mathbf{H}}}_{\alpha }},{{{\mathbf{H}}}_{D}}\} $$, where $${{{\mathbf{H}}}_{D}}$$ is the operator of the Dirichlet problem on the graph.

Inverse problems for the matrix Sturm–Liouville equation with a Bessel-type singularity

ABSTRACTThe matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems.

Inverse scattering on the line for the matrix Sturm–Liouville equation

The inverse scattering problem is studied for the matrix Sturm–Liouville equation on the line. Necessary and sufficient conditions for the scattering data are obtained.

A stable method to solve the Matrix Sturm-Liouville Equation for inhomogeneous layered structures

The study of elementary excitations in layered structures can be addressed via a Matrix Sturm-Liouville Equation of motion. A stable numerical method to solve this ubiquitous system of coupled second-order differential equations is presented. Its straightforward algorithm, especially useful for inhomogeneous layered structures, combines a local exponential approximation for the associated transfer matrix with the Hybrid matrix method. The latter is a well-known numerical stable method which is also useful to solve several boundary problems involving structures with planar, cylindrical or spherical geometry. Two numerical examples are considered and compared with the results reported previously to demonstrate the validity of the method.

Matrix Sturm–Liouville equation with a Bessel-type singularity on a finite interval

The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with the prescribed behavior at the singular point and Birkhoff-type solutions with the known asymptotics for large values of the spectral parameter. The asymptotic formulas for Stokes multipliers, connecting these two fundamental systems of solutions, are derived. We also set boundary conditions and obtain asymptotic formulas for the spectral data (the eigenvalues and the weight matrices) of the boundary value problem. Our results will be useful in the theory of direct and inverse spectral problems.

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.

Relations between Transfer Matrices and Numerical Stability Analysis to Avoid the $\Omega d$ Problem

The transfer matrix method is usually employed to study problems described by $N$ equations of matrix Sturm-Liouville (MSL) kind. In some cases a numerical degradation (the so called $\Omega d$ problem) appears thus impairing the performance of the method. We present here a procedure that can overcome this problem in the case of multilayer systems having piecewise constant coefficients. This is performed by studying the relations between the associated transfer matrix and other transfer matrix variants. In this way it was possible to obtain the matrices which can overcome the $\Omega d$ problem in the general case and then in problems which are particular cases of the general one. In this framework different strategies are put forward to solve different boundary condition problems by means of these numerically stable matrices. Numerical and analytic examples are presented to show that these stable variants are more adequate than other matrix methods to overcome the $\Omega d$ problem. Due to the ubiquity of the MSL system, these results can be applied to the study of many elementary excitations in multilayer structures.

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.

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.

Non-selfadjoint matrix Sturm–Liouville operators with spectral singularities

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm–Liouville operator L generated in L 2 ( R + , S ) by the differential expression ℓ ( y ) = - y ″ + Q ( x ) y , x ∈ R + : [ 0 , ∞ ) , and the boundary condition y ( 0 ) = 0 . Under the condition sup x ∈ R + exp ε x ‖ Q ( x ) ‖ < ∞ , ε > 0 using the uniqueness theorem of analytic functions we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities.

An inverse problem for the non-selfadjoint matrix Sturm–Liouville equation on the half-line

An inverse spectral problem is studied for the non-selfadjoint matrix Sturm–Liouville differential equation on the half-line. We give a formulation of the inverse problem, prove the corresponding uniqueness theorem and provide a constructive procedure for the solution of the inverse problem by the method of spectral mappings. The obtained results are natural generalizations of the classical results in inverse problem theory for scalar Sturm–Liouville operators.

Inverse problems for the matrix Sturm–Liouville equation on a finite interval

Inverse spectral problems are studied for the non-self-adjoint matrix Sturm–Liouville differential equation on a finite interval. We give formulations of the inverse problems, prove the corresponding uniqueness theorems, and provide a constructive procedure for the solution of the inverse problems by the method of spectral mappings. The obtained results are a natural generalization of the classical results in inverse problem theory for scalar Sturm–Liouville operators.

Matrix Riccati equations and matrix Sturm–Liouville problems

The asymptotic behavior of determinants of unitary solutions of matrix Riccati differential equations containing a large parameter is determined. The result leads to theorems on existence and asymptotic distribution of eigenvalues of indefinite matrix Sturm–Liouville problems.