Geometric characterization of nodal domains

Y. Elon, C. Joas, S. Gnutzman and U. Smilansky

Non-regular surfaces and random wave ensembles

General scope and motivation

Definitions and notations:

Separable surfaces

Nodal domains and quantum chaos:• The differences between the nodal set of classically integrable and chaotic surfaces are being under investigation for more then 25 years.

• Blum et al. (2002): counting statistics of nodal domains is

essentially different for separable and chaotic billiards; For the latter statistics follows predictions for random superpositions of plane waves (supporting a conjecture by M.V. Berry).

• Bogomolny et al. (2002): Domains statistics of random waves can be derived using a percolative model (conjecture).

Geometrical properties of nodal domains have not been studied yet, and may equip us with better understanding of the Semiclassical nature of chaotic wave patterns.

The measured statistics providesan additional test to the conjectures quoted above.

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Consider a two dimensional billiard. For the jth Eigenfunction of the laplacian:

Number the nodal domain from 1 to .

For Dk, the kth nodal domain, define the dimensionless parameter:

Where is the domain area, and is its circumference.

For a given energy interval:Define a probability measure for by:

Our interest is in the limit of high energy, where the distribution is expected to converge to a limiting .

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The derivation of for a rectangle (setting the side lengths to N,M) is a simple yet instructive example:

For all values of the quantum numbers n,m the nodal structure is of identical rectangles;The relevant ingredients are:

Approximating the discrete sum by an integral yield the limiting distribution:

This distribution has the following properties:

We are introducing analytic derivation for a family of quantum separable surfaces. The partial energies are defined by: 21

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1. compact support

2. smooth & decreasing

3. diverges as at the lower support.

4. a finite positive value at the upper support.

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Other separable surfaces

In the limit of high energy, the domains of separable surface are asymptotically rectangles. Since is a local parameter, the expression for should hold for all separable patterns, and therefore also the derived support. In addition, the minimum at ensures the square-root divergence at the lower bound, and a full analysis leads to the conclusion that all the four properties described above are universal.

Derivation of the distribution for other surfaces

Support the above conclusion. In addition for the investigated surfaces (disc and surfaces of revolution) it was shown that the distribution function can be expressed as:

Where is a slowly varying continuous decreasing function, which “measure the symmetry” between the system coordinates.

Numerical results of the distribution function for all the examined surfaces converge to the limiting distribution as , which is consistent with the order of approximation which were taken.

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Simple ↔ Multiple connectivity

Two properties of the domains seems to be relevant to the distribution of :

Convexity and curvature

Following the percolative model suggested by Bogomolny et al. , a distribution for the domains’ genus can be deduced (at least asymptotically) in addition to the area distribution. Several geometric arguments (e.g. the nodal set curvature distribution and length fluctuations, Faber-Krahn theorem in addition to other isoperimetric inequalities) considered together with results from percolation theory, leads to the assumption that distribution function should be (to the first order) a sum of two normal distributions: one for the convex domains , centered at (where is the first zero of ). The second gaussian originate from the non-convex domains.The conjectures are supported qualitatively (and for some aspects quantitatively) by preliminary numerical results, as shown.

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Correspondence to chaotic billiards

As conjectured by Berry we expect the statistics of chaotic wavefunctions to follow the predictions for random waves. Since we focus our interest in local properties, boundary effects are easy to be counted out, thus we expect a rapid convergence into the limit. The next plots take into account inner domains only, and indeed the fluctuations are weak even for low eigenvalues.

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