Publication details

Sturm-Liouville matrix differential systems with singular leading coefficient

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Year of publication 2017
Type Article in Periodical
Magazine / Source Annali di Matematica Pura ed Applicata. Series IV
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Sturm-Liouville differential equation; Linear Hamiltonian system; Generalized quasiderivative; Oscillation theory; Spectral theory; Quadratic functional; Rayleigh principle
Description In this paper we study a general even order symmetric Sturm-Liouville matrix differential equation, whose leading coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm-Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard (controllable) linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh's principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even order symmetric Sturm-Liouville matrix differential equation into the normal form. Throughout the paper we provide several examples, which illustrate our new theory.
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