Here is a toy example; I don't know how interesting this will be to physicists. The eigenvalues of the Laplacian acting on, say, smooth functions $\mathbb{R}^k/(2\pi \mathbb{Z})^k \to \mathbb{C}$ are given by
$$\{ m_1^2 + ... + m_k^2 : m_i \in \mathbb{Z} \}.$$

as a multiset (that is, with multiplicities). These are the energy eigenvalues of $n$ free non-interacting quantum particles on a circle. The multiplicity of a given eigenvalue is therefore the number of ways to write it as a sum of $k$ (integer) squares.

This is a classical number-theoretic problem. For example, it is a classical result that the number of ways to write a non-negative integer $n$ as the sum of two squares is
$$r_2(n) = 4 \sum_{d | n} \chi_4(d)$$

where $\chi_4(d)$ is equal to $0$ if $d \equiv 0, 2 \bmod 4$, equal to $1$ if $d \equiv 1 \bmod 4$, and equal to $-1$ if $d \equiv 3 \bmod 4$. In general, the number of ways $r_k(n)$ to write a non-negative integer $n$ as the sum of $k$ squares has generating function
$$\sum r_k(n) q^n = \left( \sum_{m \in \mathbb{Z}} q^{m^2} \right)^k = \theta(q)^k.$$

The function $\theta(q)$ is a theta function. Theta functions are closely related to modular forms, an important topic in number theory, and in fact the classical proof of the closed form
$$r_4(n) = 8 \sum_{d | n} [4 \nmid d]$$

(where we have used the Iverson bracket above) proceeds by showing that $\theta(q)^4$ is a modular form; see Wikipedia.

This post imported from StackExchange Physics at 2014-03-24 09:19 (UCT), posted by SE-user Qiaochu Yuan