Remarks on "Resolving isospectral `drums' by counting nodal domains"

TL;DR

This paper analytically proves that the nodal domain count uniquely distinguishes any set of four distinct positive real parameters in a family of isospectral flat 4-tori, confirming previous numerical findings.

Contribution

It provides a rigorous analytical proof that nodal counts can distinguish all parameter sets in the studied family of isospectral tori.

Findings

Nodal counts uniquely identify parameter sets

Analytical confirmation of previous numerical results

Distinguishing isospectral tori by nodal domain analysis

Abstract

In [3] the authors studied the 4-parameter family of isospectral flat 4-tori T^\pm(a,b,c,d) discovered by Conway and Sloane. With a particular method of counting nodal domains they were able to distinguish these tori (numerically) by computing the corresponding nodal sequences relative to a few explicit tuples (a,b,c,d). In this note we confirm the expectation expressed in [3] by proving analytically that their nodal count distinguishes any 4-tuple of distinct positive real numbers.