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Quantum Variance of Maass-Hecke Cusp Forms

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Abstract

In this paper we study quantum variance for the modular surface \({X=\Gamma\backslash\mathbb{H}}\), where \({\Gamma=SL_2(\mathbb{Z})}\) is the full modular group. We evaluate asymptotically the quantum variance, which is introduced by S. Zelditch and describes the fluctuations of a quantum observable. It is shown that the quantum variance is equal to the classical variance of the geodesic flow on S*X, the unit cotangent bundle of X, but twisted by the central value of the Maass-Hecke L-functions.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Peng Zhao

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  1. Peng Zhao
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Correspondence to Peng Zhao.

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Communicated by S. Zelditch

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Zhao, P. Quantum Variance of Maass-Hecke Cusp Forms. Commun. Math. Phys. 297, 475–514 (2010). https://doi.org/10.1007/s00220-009-0943-9

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