2012/06/03 by Sergio Albeverio, M. M. Malamud, Albeverio, Sergio +4 · 1 citation
Computer Science · Mathematics · #34B08 #34B20 #34B40 #47B25 #47E05 #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #math.FA #msc:34B08 #msc:34B20 #msc:34B40 #msc:47B25 #msc:47E05
paper · pdf · doi:10.48550/arxiv.1206.0479
46 pages
arxiv created 2012/06/03 · openalex publication_date 2012/06/03 · arxiv updated 2012/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study general (not necessarily Hamiltonian) first-order symmetric systems J y'(t)-B(t)y(t)=\D(t) f(t) on an interval [a,b> with the regular endpoint a. The deficiency indices n_± of the corresponding minimal relation \Tmi may be arbitrary (possibly unequal). Our approach is based on the concept of a decomposing boundary triplet, which enables one to parametrize various classes of extensions of \Tmi (self-adjoint, m-dissipative, etc.) in terms of boundary conditions imposed on regular and singular values of a function y∈ \dom \tma at the endpoints a and b respectively. In particular, we describe self-adjoint and ł-depending Nevanlinna boundary conditions which are analogs of separated ones for Hamiltonian systems. With a boundary value problem involving such conditions we associate the m-function m(\cd), which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. In the simplest case of minimal (unequal) deficiency indices n_± the m-function m(\cd) coincides with the rectangular Titchmarsh-Weyl coefficient introduced by Hinton and Schneider. We parametrize all m-functions in terms of the Nevanlinna boundary parameter at the endpoint b by means of the formula similar to the known Krein formula for resolvents. Application of these results to 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.