Boundary value problems with eigenvalue depending boundary conditions

✍ Scribed by Jussi Behrndt

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 358 KB
• Volume: 282
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200610763

✦ Synopsis

Abstract

We investigate some classes of eigenvalue dependent boundary value problems of the form
equation image
where A ⊂ A^+^ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ~0~, Γ~1~ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P~K~ (à – λ)^–1^|~K~ k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)