Sharp bounds for the intersection of nodal lines with certain curves

Jung, Junehyuk. "Sharp bounds for the intersection of nodal lines with certain curves." Journal of the European Mathematical Society 016.2 (2014): 273-288. <http://eudml.org/doc/277180>.

@article{Jung2014,
abstract = {Let $Y$ be a hyperbolic surface and let $\phi $ be a Laplacian eigenfunction having eigenvalue $-1/4-\tau ^2$ with $\tau >0$. Let $N(\phi )$ be the set of nodal lines of $\phi $. For a fixed analytic curve $\gamma $ of finite length, we study the number of intersections between $N(\phi )$ and $\gamma $ in terms of $\tau $. When $Y$ is compact and $\gamma $ a geodesic circle, or when $Y$ has finite volume and $\gamma $ is a closed horocycle, we prove that $\gamma $ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N(\phi )$ and $\gamma $ is $O(\tau )$. This bound is sharp.},
author = {Jung, Junehyuk},
journal = {Journal of the European Mathematical Society},
keywords = {nodal domain; hyperbolic surfaces; eigenfunctions; Laplacian; geodesic circle; hyperbolic surface; Laplacian; geodesic circle; eigenvalue},
language = {eng},
number = {2},
pages = {273-288},
publisher = {European Mathematical Society Publishing House},
title = {Sharp bounds for the intersection of nodal lines with certain curves},
url = {http://eudml.org/doc/277180},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Jung, Junehyuk
TI - Sharp bounds for the intersection of nodal lines with certain curves
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 2
SP - 273
EP - 288
AB - Let $Y$ be a hyperbolic surface and let $\phi $ be a Laplacian eigenfunction having eigenvalue $-1/4-\tau ^2$ with $\tau >0$. Let $N(\phi )$ be the set of nodal lines of $\phi $. For a fixed analytic curve $\gamma $ of finite length, we study the number of intersections between $N(\phi )$ and $\gamma $ in terms of $\tau $. When $Y$ is compact and $\gamma $ a geodesic circle, or when $Y$ has finite volume and $\gamma $ is a closed horocycle, we prove that $\gamma $ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N(\phi )$ and $\gamma $ is $O(\tau )$. This bound is sharp.
LA - eng
KW - nodal domain; hyperbolic surfaces; eigenfunctions; Laplacian; geodesic circle; hyperbolic surface; Laplacian; geodesic circle; eigenvalue
UR - http://eudml.org/doc/277180
ER -