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arXiv · 2411.02436

Degeneracies In a Weighted Sum of Two Squares

Ishan Vinayagam Ramesh·Maxim Olshanii
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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.

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Ishan Vinayagam Ramesh, Maxim Olshanii. 2024-11-01. Degeneracies In a Weighted Sum of Two Squares. https://arxiv.org/abs/2411.02436

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