Compressed resolvents and reduction of spectral problems on star graphs

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, ẽ₁ 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?