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Nodal Domains of Maass Forms I

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Abstract

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the L 2-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex L ∞-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Amit Ghosh

  2. Department of Mathematics, Bar-Ilan University, 52900, Ramat Gan, Israel

    Andre Reznikov

  3. School of Mathematics, Institute for Advanced Study, Princeton, USA

    Peter Sarnak

  4. Department of Mathematics, Princeton University, Princeton, NJ, 08540, USA

    Peter Sarnak

Authors
  1. Amit Ghosh
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  2. Andre Reznikov
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  3. Peter Sarnak
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Correspondence to Peter Sarnak.

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Dedicated belatedly to Dennis Hejhal on the occasion of his sixtieth birthday

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Ghosh, A., Reznikov, A. & Sarnak, P. Nodal Domains of Maass Forms I. Geom. Funct. Anal. 23, 1515–1568 (2013). https://doi.org/10.1007/s00039-013-0237-4

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  • DOI: https://doi.org/10.1007/s00039-013-0237-4

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