Trace formulae for graph Laplacians with applications to recovering matching conditions. (English) Zbl 1289.34088

The authors consider graph Laplacians on finite compact metric graphs under the assumption that the matching conditions are of generalized \(\delta\) or \(\delta'\) type (with certain coefficients like in the classical third boundary condition). Using the boundary triples approach, they study spectral properties of these operators. In each of the two cases, they find an infinite series of trace formulas linking two different graph Laplacians with coinciding spectra. This result is applied to the problem of reconstructing the matching conditions to the given spectrum. The uniqueness in this problem is proved in some cases.


34B45 Boundary value problems on graphs and networks for ordinary differential equations
47A75 Eigenvalue problems for linear operators
34A55 Inverse problems involving ordinary differential equations
34L05 General spectral theory of ordinary differential operators
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