Boundary Value Problems with Local Generalized Nevanlinna Functions in the Boundary Condition

Abstract

For a class of abstract λ-dependent boundary value problems where a local variant of generalized Nevanlinna functions appears in the boundary condition, linearizations are constructed and their local spectral properties are investigated. Mathematics Subject Classification (2000). Primary 47B50, 34B07; Secondary 46C20, 47A06, 47B40. Keywords. Boundary value problems, symmetric and selfadjoint operators and relations in Krein spaces, generalized Nevanlinna functions, boundary value spaces, Weyl functions.

Boundary relations and their Weyl families

“…The general case will be carried out in a forthcoming paper. Recently the coupling method for the construction of generalized resolvents in [12] has been applied by J. Behrndt and P. Jonas [4]. Their treatment involves a scalar function τ (·), belonging to the class of local generalized Nevanlinna functions, which generates a selfadjoint exit space extension in a Kreȋn space.…”

Sturm–Liouville operators with indefinite weight functions and eigenvalue depending boundary conditions

“…For a positive function r and a generalized Nevanlinna function boundary value problems of the form (2)- (3) have been studied in a more or less abstract framework extensively in the last decades (see e.g. [4,12,18,21,31,33] and the references quoted in [17]).…”

“…In order to solve (2)- (3) in Section 4 we investigate the abstract -dependent boundary value problem…”

Locally Definitizable Operators: The Local Structure of the Spectrum

“…For locally definitizable operators in the context of (indefinite) Sturm-Liouville problems we refer to [15,20,27,64], for λ-dependent boundary value problems see [16,19,63] in the context of PT -symmetric operators see [10,11], in the study of partial differential equations [17,29], for a special form of the Krein-Naimark formula see [13,23] and in the study of problems of Klein-Gordon type see [56,58]. [67] and to the monographs [3,35].…”