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A trace formula for the nodal count sequence

Towards counting the shape of separable drums

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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.

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Authors and Affiliations

  1. Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot, 76100, Israel

    S. Gnutzmann & U. Smilansky

  2. Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195, Berlin, Germany

    S. Gnutzmann

  3. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

    S. Gnutzmann

  4. School of Mathematics, Bristol University, Bristol, BS8 1TW, UK

    P. Karageorge & U. Smilansky

Authors
  1. S. Gnutzmann
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  2. P. Karageorge
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  3. U. Smilansky
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Correspondence to S. Gnutzmann.

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Gnutzmann, S., Karageorge, P. & Smilansky, U. A trace formula for the nodal count sequence. Eur. Phys. J. Spec. Top. 145, 217–229 (2007). https://doi.org/10.1140/epjst/e2007-00158-y

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  • DOI: https://doi.org/10.1140/epjst/e2007-00158-y

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