Spectral Theory for Second-Order Vector Equations on Finite Time-Varying Domains

TL;DR

This paper develops spectral theory for second-order vector dynamic equations on finite, non-uniform domains, introducing self-adjointness concepts and extending results to higher-order and Hamiltonian systems on time scales.

Contribution

It introduces a new framework for spectral analysis of second-order vector dynamic equations with singular coefficients on finite, variable domains, including self-adjointness and dual orthogonality.

Findings

Established fundamental spectral results for the equations.

Proved dual orthogonality of eigenfunctions in a special case.

Extended the theory to higher-order and Hamiltonian systems.

Abstract

In this study, we are concerned with spectral problems of second-order vector dynamic equations with two-point boundary value conditions and mixed derivatives, where the matrix-valued coefficient of the leading term may be singular, and the domain is non-uniform but finite. A concept of self-adjointness of the boundary conditions is introduced. The self-adjointness of the corresponding dynamic operator is discussed on a suitable admissible function space, and fundamental spectral results are obtained. The dual orthogonality of eigenfunctions is shown in a special case. Extensions to even-order Sturm-Liouville dynamic equations, linear Hamiltonian and symplectic nabla systems on general time scales are also discussed.