Semiclassical analysis of distinct square partitions

TL;DR

This paper analyzes the number of partitions of integers into distinct squares using semiclassical methods, deriving asymptotic formulas, spectral insights, and a trace formula that captures oscillations for large n.

Contribution

It introduces a semiclassical trace formula for $P(n)$, providing higher-order asymptotics and linking oscillations to Pythagorean triples, advancing understanding of partition functions.

Findings

Derived an integral representation of $P(n)$.

Obtained higher-order asymptotic expressions for $P(n)$.

Developed a semiclassical trace formula accurately describing oscillations for large $n$.

Abstract

We study the number $P(n)$ of partitions of an integer $n$ into sums of distinct squares and derive an integral representation of the function $P(n)$. Using semi-classical and quantum statistical methods, we determine its asymptotic average part $P_{as}(n)$, deriving higher-order contributions to the known leading-order expression [M. Tran {\it et al.}, Ann.\ Phys.\ (N.Y.) {\bf 311}, 204 (2004)], which yield a faster convergence to the average values of the exact $P(n)$. From the Fourier spectrum of $P(n)$ we obtain hints that integer-valued frequencies belonging to the smallest Pythagorean triples $(m,p,q)$ of integers with $m^2+p^2=q^2$ play an important role in the oscillations of $P(n)$. Finally we analyze the oscillating part $\delta P(n)=P(n)-P_{as}(n)$ in the spirit of semi-classical periodic orbit theory [M. Brack and R. K. Bhaduri: {\it Semiclassical Physics} (Bolder, Westview Press, 2003)]. A semi-classical trace formula is derived which accurately reproduces the exact $\delta P(n)$ for $n > \sim 500$ using 10 pairs of `orbits'. For $n > \sim 4000$ only two pairs of orbits with the frequencies 4 and 5 -- belonging to the lowest Pythagorean triple (3,4,5) -- are relevant and create the prominent beating pattern in the oscillations. For $n > \sim 100,000$ the beat fades away and the oscillations are given by just one pair of orbits with frequency 4.