Degeneracies In a Weighted Sum of Two Squares

TL;DR

This work analyzes degeneracies in the energy spectrum of a 1:√3 rectangular billiard, where the energy is $\mathcal{E}_{n_1,n_2}=3n_1^2+n_2^2$. It shows that degenerate levels fall into two parity-based classes: same-parity levels hosting Perrin-type 3-fold degeneracies and opposite-parity levels hosting Brahmagupta-type 2-fold degeneracies, with all pairs of degenerate states expressible via Brahmagupta constructions. The authors provide both a rigorous parity-based framework and numerical validation up to $\mathcal{E}_{\max}=2700$, identifying first instances at $\mathcal{E}=28$ (Perrin) and $\mathcal{E}=91$ (Brahmagupta), including half-integer parameter cases. The results offer a structured view of degeneracies beyond conventional symmetry arguments and point to deeper connections with tiling symmetries and number-theoretic representations.

Abstract

This work is an attempt to classify and quantify instances when a weighted sum of two squares of positive integers, $3n_{1}^2+n_{2}^2$, can be realized in more than one way. Our project was inspired by a particular study of two-dimensional quantum billiards [S. G. Jackson, H. Perrin, G. E. Astrakharchik, and M. Olshanii, SciPost Phys. Core 7, 062 (2024)] where the weighted sums of interest represents an energy level with the two integers being the billiard's quantum numbers; there, the 3-fold degeneracies seem to dominate the energy spectrum. Interestingly, contrary to the conventional paradigm, these degeneracies are not caused by some non-commuting symmetries of the system.