Boundary pairs and boundary conditions for general (not necessarily definite) first‐order symmetric systems with arbitrary deficiency indices

Abstract

AbstractWe investigate in the paper general (not necessarily definite) first‐order symmetric systems of differential equations in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary pair \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \mathcal H_0\oplus \mathcal H_1, \Gamma \rbrace$\end{document} for A*, where A is a symmetric linear relation, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_0$\end{document} is a boundary Hilbert space, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_1$\end{document} is a subspace in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_0$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma :A^*\rightarrow \mathcal H_0\oplus \mathcal H_1$\end{document} is a (possibly multivalued) linear mapping satisfying the abstract Green's identity. Unlike known concept of a boundary pair for A* our definition is suitable for relations A with possibly unequal deficiency indices n±(A).Next, the general symmetric system Jy′(t) − B(t)y(t) = Δ(t)f(t) on an interval \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal I=(a,b),\; -\infty \le a < b\le \infty ,$\end{document} is considered in the paper. We characterize explicitly the corresponding minimal relation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {min}}$\end{document}, which enables us to construct a special (so‐called decomposing) boundary pair \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \mathcal H_0\oplus \mathcal H_1, \Gamma \rbrace$\end{document} for the maximal relation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {max}}$\end{document}. It turns out that the system is definite if and only if the mapping Γ is single‐valued, in which case a decomposing boundary pair turns into the decomposing boundary triplet \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Pi =\lbrace \mathcal H,\Gamma _0,\Gamma _1\rbrace$\end{document} for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {max}}$\end{document}. By using such a triplet we describe in terms of boundary conditions proper extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {min}}$\end{document} in the case of the regular endpoint a and arbitrary (possibly unequal) deficiency indices \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$n_\pm (T_{\textit {min}})$\end{document}. We also show that self‐adjoint decomposing boundary conditions exist only for Hamiltonian systems; moreover, we describe all such conditions in the compact form. These results are generalizations of the known results by Rofe‐Beketov on regular differential operators. Finally, we characterize all maximal dissipative and accumulative separated boundary conditions for arbitrary (not necessarily Hamiltonian) definite symmetric systems