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Monatshefte für Mathematik

, Volume 183, Issue 2, pp 311–328

First Online: 31 October 2016Received: 02 March 2016Accepted: 17 October 2016DOI: 10.1007-s00605-016-1001-2

Cite this article as: Maffucci, R.W. Monatsh Math 2017 183: 311. doi:10.1007-s00605-016-1001-2

Abstract

We consider random Gaussian eigenfunctions of the Laplacian on the standard torus, and investigate the number of nodal intersections against a line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on whether the slope of the straight line is rational or irrational. Our findings exhibit a close relation between this problem and the theory of lattice points on circles.

KeywordsNodal intersections Arithmetic random waves Gaussian eigenfunctions Lattice points on circles Communicated by A. Constantin.

Mathematics Subject Classification11P21 60G15 



Autor: Riccardo W. Maffucci

Fuente: https://link.springer.com/







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