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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, 3n2 1 +n2 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.

Author Bio

Ishan V. Ramesh is currently a high school senior at Hopkinton High School, Hopkinton, MA. From 2021 to 2024, he simultaneously participated in national quiz bowl tournaments and FRC Robotics competitions. The latter resulted in him winning a Rookie All-Star Award for the state of Virginia. More recently, Ramesh has devoted significant portions of his research portfolio to the study of classes of systematic representations (or degeneracies) of quantum billiards. Ramesh devotes much of his remaining free time to composing prose poetry and travel photography.

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