Spectral theory for a singular linear system of Hamiltonian differential equations

Abstract

In 1910 Weyl [9] inaugurated the modern theory of singular self-adjoint differential operators, considering, in particular, second-order differential operators with real coefficients. Since then, his results have been generalized by various authors to the case of arbitrary even order self-adjoint differential equations with real coefficients. Also, in 1954 Coddington [2] treated the case of an arbitrary n-th order self-adjoint differential equation with complex coefficients, He obtained the Parseval equality and spectral expansion associated with the singular case directly, through the consideration of such a problem as a limiting case of corresponding self-adjoint two-point boundary-value problems on compact subintervals of the reals. Using the same general concept, Coddington and Levinson [3] derived a Green’s function, which in turn enabled them to obtain the spectral matrix for a singular problem involving a linear homogeneous n-th order differential operator, and then proceeded to derive the Parseval equality and spectral expansions. Recently Brauer [I] treated similar problems which involve a definitely self-adjoint vector differential operator, of the sort that has been treated by Reid [6] and others. He obtained the Green’s matrix and spectral matrix through the use of the spectral theorem and the theory of direct integrals. The main purpose of the present paper is to employ the general method