Nodal Domains of Maass Forms I

Abstract

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.

Quantum unique ergodicity and the number of nodal domains of eigenfunctions

“…Constant lower bound for L 2 norm of the restriction of an eigenfunction to a geodesic segment is first proven in [GRS13], from the arithmetic QUE theorem [Lin06,Sou10].…”

“…Graph structure of the nodal set and Euler's inequality. In this section we briefly review the topological argument in [GRS13,JZ13] on bounding the number of nodal domains from below by the number of zeros on Fix(τ ). We refer the readers to [JZ13] for details.…”

“…We now use the topological argument in [GRS13,JZ13] to conclude an analogy of Theorem 1.5 for odd eigenfunctions.…”

“…the number of singular points) of the eigenfunction along Fix(τ ). Such an argument is first developed in [GRS13] and we review in §4.1 in terms of the nodal graphs and Euler's inequality as in [JZ13]. We then use Bochner's theorem and a Rellich type identity to deduce from QUE that even (resp.…”

“…(See [GRS13,Jun13,BR13,JZ13,Mag13,GRS14], where such an idea is used to prove a lower bound for the number of sign changes in various contexts.) In order to bound u n L 1 (β) from below using Hölder's inequality, the authors use the Quantum Ergodic Restriction (QER) theorem [TZ13,DZ13] for the lower bound of u n L 2 (β) and the point-wise Weyl law with an improved error term [Bér77] for the upper bound of u n L ∞ (β) .…”