Boundary relations and boundary conditions for general (not necessarily definite) canonical systems with possibly unequal deficiency indices

TL;DR

This paper develops a generalized boundary relation framework for canonical systems with possibly unequal deficiency indices, extending classical results and characterizing boundary conditions for various types of systems.

Contribution

Introduces a new boundary relation concept applicable to relations with unequal deficiency indices and generalizes boundary condition characterizations for canonical systems.

Findings

Boundary relation $\\G$ applicable to relations with unequal deficiency indices.

Characterization of self-adjoint boundary conditions for Hamiltonian systems.

Description of maximal dissipative and accumulative boundary conditions for canonical systems.

Abstract

We investigate in the paper general (not necessarily definite) canonical systems of differential equation in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary relation $\G:\gH^2\to\HH$ for $A^*$, where $\gH$ is a Hilbert space, $A$ is a symmetric linear relation in $\gH, \cH_0$ is a boundary Hilbert space and $\cH_1$ is a subspace in $\cH_0$. Unlike known concept of a boundary relation (boundary triplet) for $A^*$ our definition of $\G$ is applicable to relations $A$ with possibly unequal deficiency indices $n_\pm(A)$. Next we develop the known results on minimal and maximal relations induced by the general canonical system $ J y'(t)-B(t)y(t)=\D (t)f(t)$ on an interval $\cI=(a,b),\; -\infty\leq a<b\leq\infty $ and then by using a special (so called decomposing) boundary relation for $\Tma$ we describe in terms of boundary conditions proper extensions of $\Tmi$ in the case of the regular endpoint $a$ and arbitrary (possibly unequal) deficiency indices $n_\pm (\Tmi)$. If the system is definite, then decomposing boundary relation $\G$ turns into the decomposing boundary triplet $\Pi=\bt$ for $\Tma$. Using such a triplet we 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. We characterize also all maximal dissipative and accumulative separated boundary conditions, which exist for arbitrary (not necessarily Hamiltonian) definite canonical systems.