It is proven that the spectrum of the Laplacian on a metric graph $\Gamma$ contains arithmetic sequences if and only if the graph has a loop - an edge connected to one vertex by both end points. Moreover the length of the longest possible arithmetic subsequence is estimated using the corresponding discrete graph $G$.
Our main tool is diophantine analysis, specifically "Lang's $G_m$ Conjecture" concerning the intersection of the division group of a finitely generated subgroup of $(C^ * )^ N$ with a subvariety of $(C^ * )^ N$. On our way we prove recent Colin de Verdière's Conjecture concerning structure of polynomials associated with metric graphs.
The trace formula connecting spectra of standard Laplacians on metric graphs to the sets of periodic orbits allows us to construct a large family of exotic crystalline measures, studied recently by Y. Meyer. Crystalline measures are discrete measures with Fourier transform being a discrete measure as well. Our analysis in the first part imply that constructed measures are not just combinations of Poisson summation formulae.
This is a joint work with Peter Sarnak.
This talk is part of series of talks affiliated with the virtual Mittag-Leffler workshop "Scattering, microlocal analysis and renormalization", organized by Claudio Dappiaggi, Jacob Schach MÃ¸ller and Michal Wrochna. The full schedule can be found at: