Eur. Phys. J. Special Topics 145, 217-229 (2007)
https://doi.org/10.1140/epjst/e2007-00158-y

A trace formula for the nodal count sequence

Towards counting the shape of separable drums

¹ Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, 76100, Israel
² Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
³ School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
⁴ School of Mathematics, Bristol University, Bristol, BS8 1TW, UK

Corresponding authors: sven.gnutzmann@nottingham.ac.uk panos.karageorge@bristol.ac.uk uzy.smilansky@weizmann.ac.il

Abstract

The sequence of nodal count is considered for separable drums. A recently derived trace formula for this sequence stores geometrical information of the drum. This statement is demonstrated in detail for the Laplace-Beltrami operator on simple tori and surfaces of revolution. The trace formula expresses the cumulative sum of nodal counts This sequence is expressed as a sum of two parts: a smooth (Weyl like) part which depends on global geometrical parameters, and a fluctuating part which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical context of the nodal sequence is thus explicitly revealed.