The Interaction Matrix of a Reflection-Symmetric Periodic Pair Interaction is Symmetric and Diagonal in the Trigonometric Basis, with Eigenvalues Its Symbol

In the settings of The Real Numbers: Standing Notation and Background and The Flat Torus: Standing Notation, let $M\in\mathbb{N}$, and let $L$, $\Gamma_{M}$, $W_{L}$, $\psi_{k}$, the lattice torus $\mathbb{L}_{M}$, lattice fields and their discrete Fourier coefficients $\hat{f}(k)$ be those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field. Let $\mathfrak{j}$ be a reflection-symmetric periodic pair interaction of side $L$, with interaction matrix $J$ and symbol $\hat{J}$. Sums over finite index sets are those of Sum over a Finite Index Set, $L^{n}$ is the natural power of $L$ in $\mathbb{R}$, and $|s|$ is the absolute value of $s\in\mathbb{R}$.

1. (Symmetry)¶ $J(y,z)=J(z,y)$ for all $y,z\in\mathbb{L}_{M}$.

2. (The trigonometric maps are eigenvectors)¶ For every $k\in\Gamma_{M}$ and every $y\in\mathbb{L}_{M}$,

$$\sum_{z\in\mathbb{L}_{M}}J(y,z)\,\psi_{k}(z)=\hat{J}(k)\,\psi_{k}(y).$$

3. (The interaction energy)¶ For all lattice fields $f$ and $g$,

$$\sum_{y\in\mathbb{L}_{M}}\sum_{z\in\mathbb{L}_{M}}J(y,z)\,f(y)\,g(z)=L^{n}\sum_{k\in\Gamma_{M}}\hat{J}(k)\,\hat{f}(k)\,\hat{g}(k).$$

4. (Bound on the symbol)¶ For every $k\in\Gamma_{M}$, $|\hat{J}(k)|\le\sum_{w\in W_{L}}|\mathfrak{j}(w)|$.