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 formula is reminiscent of the original Gelfand-Levitan result on the segment with Neumann boundary 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.
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This is joint work with prof. Pedro Freitas.
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[1] P. Freitas, J. Lipovsky, A Gelfand-Levitan trace formula for generic quantum graphs, Anal. Math. Phys. 11 (2021), 56 [mp_arc 19-4; arXiv: 1901.07790 [math-ph]]