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.