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    Title      l  Quantitative quantum ergodicity and the nodal domains of Maass-Hecke cusp forms
    Author   l  Junehyuk Jung
    Date   l  2014-01-10
    Category   l  ASARC-Preprint-14-01
    Download    l 1389574417_0.794006.pdf  
We prove a quantitative statement of the quantum ergodicity for Maass-Hecke cusp forms on $SL(2,mathbb{Z})ackslash mathbb{H}$. As an application of our result, we obtain a sharp lower bound for the $L^2$-norm of the restriction of even Maass-Hecke cusp form $f$s to any fixed compact geodesic segment in ${iy~|~y>0} subset mathbb{H}$, with a possible exceptional set which is polynomially smaller in the size than the set of all $f$. We also improve $L^infty$ estimate for Maass-Hecke cusp forms given by Iwaniec and Sarnak, for almost all Maass-Hecke cusp forms. We then deduce that the number of nodal domains of $f$ which intersect a fixed geodesic segment increases with the eigenvalue, with a small number of exceptional $f$s. In the recent work of Ghosh, Reznikov, and Sarnak, they prove the same statement for all $f$ without exception, assuming the Lindelof Hypothesis and that the geodesic segment is long enough. For almost all Maass-Hecke cusp forms, we give better lower bound of number of nodal domains.

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