Remarks on nodal volume statistics for regular and chaotic wave functions in various dimensions

TL;DR

This paper analyzes the statistical properties of nodal set volumes for wave functions in regular and chaotic systems across various dimensions, providing explicit mean, variance, and distribution results, and exploring boundary effects.

Contribution

It offers explicit formulas for nodal volume statistics in regular and chaotic wave functions, including boundary effects, and proposes conjectures on their generalization to broader shapes.

Findings

Explicit mean and variance formulas for nodal volume in arbitrary dimensions.

Boundary effects reduce nodal volume in regular shapes and are calculable.

Universal features distinguish regular and chaotic wave functions' nodal statistics.

Abstract

We discuss the statistical properties of the volume of the nodal set of wave function for two paradigmatic model systems which we consider in arbitrary dimension $s\ge 2$: the cuboid as a paradigm for a regular shape with separable wave functions, planar random waves as an established model for chaotic wave functions in irregular shapes. We give explicit results for the mean and variance of the nodal volume in arbitrary dimension, and for their limiting distribution. For the mean nodal volume we calculate the effect of the boundary of the cuboid where Dirichlet boundary conditions reduce the nodal volume compared to the bulk. Boundary effects for chaotic wave functions are calculated using random waves which satisfy a Dirichlet boundary condition on a hyperplane. We put forward several conjectures what properties of cuboids generalise to general regular shapes with separable wave functions and what properties of random waves can be expected for general irregular shapes. These universal features clearly distinct between the two cases.