Title: Compressed Resolvents and Reduction of Spectral Problems on Star Graphs
Language: Deutsch
Authors: Brown, Basil Malcolm 
Langer, Heinz
Tretter, Christiane 
Category: Research Article
Keywords: Compressed resolvent; Differential operator; Star graph; Path graph; Canonical system; Krein string; Stieltjes string
Issue Date: 2019
Journal: Complex Analysis and Operator Theory
In this paper a two-step reduction method for spectral problems on a star graph with n+1 edges e0,e1,…,en and a self-adjoint matching condition at the central vertex v is established. The first step is a reduction to the problem on the single edge e0 but with an energy depending boundary condition at v. In the second step, by means of an abstract inverse result for Q-functions, a reduction to a problem on a path graph with two edges e0, e˜1 joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realizations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer e.g. the following question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type?
DOI: 10.1007/s11785-018-0793-6
Library ID: AC15368599
URN: urn:nbn:at:at-ubtuw:3-5689
ISSN: 1661-8254
Organisation: E101 - Institut für Analysis und Scientific Computing 
Publication Type: Article
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