A new proof of the Voronoï summation formula

We present a short alternative proof of the Voronoï summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schrödinger operator on a non-compact quantum graph \mathfrak{G} [S. Egger né Endres and F. Steiner, J. Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new proof of a non-trivial identity for a particular Lambert series which involves the divisor function d(n) and is identical with the trace of the Euclidean wave group of the Laplacian on the infinite graph \mathfrak{G}.

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