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

Abstract

We discuss the statistical properties of the volume of the nodal set of wave functions for two paradigmatic model systems which we consider in arbitrary dimension s ≥2: the cuboid as a paradigm for a regular shape with separable wave functions and 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 the 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 with 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 on what properties of cuboids generalize 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 distinguish between the two cases.