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

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.