We generalize the result given by Gelfand and Levitan for the Schroedinger operator on a segment with Neumann coupling condition. We give a trace formula for the quantum graph with arbitrary edge lengths and generic coupling conditions. The only case of coupling conditions which is excluded is the condition with the unitary coupling matrix having eigenvalue -1 (hence it is a set of measure zero in the set of all self-adjoint couplings). However, the considered set does not include Dirichlet, standard or delta-conditions.

This is joint work with Pedro Freitas.

Financiado pela FCT no âmbito dos projetos IF/00069/2015 e PTDC/MAT-PUR/30234/2017.