JournalsjemsVol. 16, No. 2pp. 273–288

Sharp bounds for the intersection of nodal lines with certain curves

  • Junehyuk Jung

    Princeton University, USA
Sharp bounds for the intersection of nodal lines with certain curves cover
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Abstract

Let YY be a hyperbolic surface and let ϕ\phi be a Laplacian eigenfunction having eigenvalue 1/4τ2-1/4-\tau^2 with τ>0\tau>0. Let N(ϕ)N(\phi) be the set of nodal lines of ϕ\phi. For a fixed analytic curve γ\gamma of finite length, we study the number of intersections between N(ϕ)N(\phi) and γ\gamma in terms of τ\tau. When YY is compact and γ\gamma a geodesic circle, or when YY has finite volume and γ\gamma is a closed horocycle, we prove that γ\gamma is ``good'' in the sense of \cite{pa}. As a result, we obtain that the number of intersections between N(ϕ)N(\phi) and γ\gamma is O(τ)O(\tau). This bound is sharp.

Cite this article

Junehyuk Jung, Sharp bounds for the intersection of nodal lines with certain curves. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 273–288

DOI 10.4171/JEMS/433