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

A trace formula for the nodal count sequence

Towards counting the shape of separable drums

S. Gnutzmann^{1, 2, 3}, P. Karageorge⁴ and U. Smilansky<sup>1, 4

1</sup>  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

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

(Published online: 26 June 2007)

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.

© EDP Sciences, Springer-Verlag 2007