On Eigenfunction Expansions of First-Order Symmetric Systems and Ordinary Differential Operators of an Odd Order

Abstract

We study general (not necessarily Hamiltonian) first-order symmetric systems ${J y'-B(t)y=\Delta(t) f(t)}$ on an interval ${\mathcal{I}=[a,b \rangle}$ with the regular endpoint a. It is assumed that the deficiency indices ${n_{\pm}(T_{\rm min})}$ of the minimal relation ${T_{\rm min}}$ satisfy ${n_{+}(T_{\rm min})\leq n_{-}(T_{\rm min})}$. We define special ${\lambda}$-depending boundary conditions with Nevanlinna-type spectral parameter ${\tau=\tau(\lambda)}$ at the singular endpoint b. With boundary value problem involving such conditions, we associate an exit space self-adjoint extension ${\tilde{T}}$ of ${T_{\rm min}}$ and the ${N \times N}$-matrix m-function ${m(\cdot)}$ of the size ${N={\rm dim ker} (iJ+I)}$. The role of ${m(\cdot)}$ is similar to that of the Weyl function for a Hamiltonian system. Using the m-function, we obtain the eigenfunction expansion with ${N \times N}$-matrix spectral function ${\Sigma(\cdot)}$. Moreover, we represent ${\Sigma(\cdot)}$ immediately in terms of a boundary parameter ${\tau}$. We also characterize certain spectral properties of the extension ${\tilde{T}}$. Application of these results to ordinary differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh–Weyl theory of such operators.