Eigenvalue Cluster Traces for Quantum Graphs with Equal Edge Lengths

On a finite metric graph with standard vertex conditions and equal edge lengths, the large magnitude eigenvalues of the Schrödinger operator ∆ + q cluster near the eigenvalues of the Laplace operator ∆. Based on the spectral ’periodicity’ of ∆, the clusters can be partitioned into a finite collection of classes. There is a class dependent formula for the cluster trace (or average eigenvalue shift) in the large magnitude limit which expresses the trace as a function of the edge integrals ∫ e q and data from the underlying combinatorial graph. 2000 Mathematics Subject Classification 34B45