Geometric characterization of nodal domains: the area-to-perimeter ratio

Abstract

In an attempt to characterize the distribution of forms and shapes of nodal domains in wave functions, we define a geometric parameter - the ratio $\rho$ between the area of a domain and its perimeter, measured in units of the wavelength $1/\sqrt{E}$. We show that the distribution function $P(\rho )$ can distinguish between domains in which the classical dynamics is regular or chaotic. For separable surfaces, we compute the limiting distribution, and show that it is supported by an interval, which is independent of the properties of the surface. In systems which are chaotic, or in random-waves, the area-to-perimeter distribution has substantially different features which we study numerically. We compare the features of the distribution for chaotic wave functions with the predictions of the percolation model to find agreement, but only for nodal domains which are big with respect to the wavelength scale. This work is also closely related to, and provides a new point of view on isoperimetric inequalities.

22 pages, 11 figures

References in corpus (3)

• SLE description of the nodal lines of random wave functions
• Nodal Domain Statistics for Quantum Maps, Percolation and SLE
• Geometric characterization of nodal domains: the area-to-perimeter ratio

Cited by in corpus (7)

• Eigenvectors of the discrete Laplacian on regular graphs - a statistical approach
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• Gaussian Waves on the Regular Tree
• Geometric characterization of nodal domains: the area-to-perimeter ratio
• Eigenfunctions and Nodal Sets
• Neumann Domains on Quantum Graphs
• Level sets percolation on chaotic graphs