Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures

Abstract

We consider a class of self-adjoint extensions using the boundary triplet technique. Assuming that the associated Weyl function has the special form M(z)=(m(z)Id−T)n(z)−1 with a bounded self-adjoint operator T and scalar functions m,n we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for T. As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete Laplacians.