Geometric Characterization of Nodal Patterns and Domains Y. Elon, S. Gnutzman, C. Joas U. Smilansky

Introduction 2002 – Blum, Gnutzman and Smilansky: a “Chaotic billiards can be discerned by counting nodal domains.”

Research goal – Characterizing billiards by investigating geometrical features of the nodal domains: Helmholtz equation on 2d surface (Dirichlet Boundary conditions): - the total number of nodal domains of.

Consider the dimensionless parameter:

For an energy interval:, define a distribution function:

Is there a limiting distribution? What can we tell about the distribution?

Rectangle

1.Compact support: 2. Continuous and differentiable 3. 4.

Rectangle the geometry of the wave function is determined by the energy partition between the two degrees of freedom.

Rectangle can be determined by the classical trajectory alone. Action-angle variables:

Disc the nodal lines were estimated using SC method, neglecting terms of order.

n’=1 n’=4 n’=3 n’=2

RectangleDisc Same universal features for the two surfaces:

Disc 1.Compact support: 2. Continuous and differentiable n=1 m=o

Surfaces of revolution Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve). Same approximations were taken as for the Disc.

Surfaces of revolution Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve). Same approximations were taken as for the Disc.

n’=1 n’=2 n’=3 n’=4

For the Disc: For a surface of revolution:

Following those notations:

“Classical Calculation”:

1.Look at (Classical Trajectory)

“Classical Calculation”: 2. Find a point along the trajectory for which:

“Classical Calculation”: 3. Calculate

Separable surfaces 2. can be deduced (in the SC limit) knowing the classical trajectory solely. 1.In the SC limit, has a smooth limiting distribution with the universal characteristics: - Same compact support: - diverge like at the lower support - go to finite positive value at the upper support

Random waves

Two properties of the Nodal Domains were investigated: 1.Geometrical: 2. Topological: genus – or: how many holes? G=0 G=2 G=1

Random waves

Model: ellipses with equally distributed eccentricity and area in the interval:

Random waves d

Genus The genus distributes as a power law!

Genus In order to find a limiting power law – check it on the sphere

Genus Power law? Saturation? Fisher’s exp:

Random waves 1.The distribution function has different features for separable billiards and for random waves. 2.The topological structure (i.e. genus distribution) shows complicate behavior – decays (at most) as a power-low.

Open questions: Connection between classical Trajectories and. Analytic derivation of for random waves. Statistical derivation of the genus distribution Chaotic billiards.