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

Each result cited is universally quantified over the data in its own statement. Points of $\mathbb{R}^{n}$, and of $\mathbb{Z}^{n}\subseteq\mathbb{R}^{n}$ (Lattice-Periodic Functions and the Periodic Function Classes §lattice), are added, subtracted and multiplied by real numbers componentwise (Scalar Multiple of a Point of $\mathbb{R}^n$). The side $L$ is a natural number, read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers, so $L\in\mathbb{Z}$ by Arithmetic, Order, Discreteness and Intervals of the Integers §sign; and $0,1\in\mathbb{Z}$ and $\mathbb{Z}$ is closed under $+$, $-$ and multiplication by Arithmetic, Order, Discreteness and Intervals of the Integers §ring, so that sums, differences and integer multiples of points of $\mathbb{Z}^{n}$ lie in $\mathbb{Z}^{n}$. For $m\in\mathbb{Z}$, $C_{m}$ and $S_{m}$ are the trigonometric monomials $C_{m}(t)=\cos(2\pi mt)$ and $S_{m}(t)=\sin(2\pi mt)$ of Cell Integrals of the Trigonometric Monomials, and $\phi_{m}$ is the one-dimensional map of The Trigonometric System on the Torus §one-dimensional; since $\psi_{k}$ is the map written $e_{k}$ in The Trigonometric System on the Torus §system (The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field),

$$\psi_{k}(x)=\prod_{i=1}^{n}\phi_{k_{i}}(x_{i})\qquad(k\in\mathbb{Z}^{n},\ x\in\mathbb{R}^{n}).$$

The map $w\mapsto L^{-1}w$ from $W_{L}$ to $\mathbb{L}_{M}$ is a bijection: it is onto by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §lattice, and $L^{-1}w=L^{-1}w'$ implies $w=w'$ after multiplying each component by $L$. Every site $y$ is therefore $L^{-1}w$ for exactly one $w\in W_{L}$, and by Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols §matrix, $J(L^{-1}w,L^{-1}w')=\mathfrak{j}(w-w')$ for $w,w'\in W_{L}$.

Step 1: symmetry (claim 1). For $p\in[n]$ the reflection $R_{p}$ of Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols maps $\mathbb{Z}^{n}$ to $\mathbb{Z}^{n}$. Let $b\in\mathbb{Z}^{n}$. Put $b^{(0)}=b$ and $b^{(p)}=R_{p}b^{(p-1)}\in\mathbb{Z}^{n}$ for $p\in[n]$. By induction on $p$, the $i$-th component of $b^{(p)}$ is $-b_{i}$ for $i\le p$ and $b_{i}$ for $p<i$, so $b^{(n)}=-b$; and $\mathfrak{j}(b^{(p)})=\mathfrak{j}(R_{p}b^{(p-1)})=\mathfrak{j}(b^{(p-1)})$ by Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols §interaction. Hence $\mathfrak{j}(-b)=\mathfrak{j}(b)$. Now let $y=L^{-1}w$ and $z=L^{-1}w'$ with $w,w'\in W_{L}$. Then $J(z,y)=\mathfrak{j}(w'-w)=\mathfrak{j}(-(w-w'))=\mathfrak{j}(w-w')=J(y,z)$.

Step 2: two involutions of $W_{L}$. A map $\tau:W_{L}\to W_{L}$ with $\tau(\tau(u))=u$ for every $u\in W_{L}$ is a bijection: for $v\in W_{L}$ one has $\tau(\tau(v))=v$, and $\tau(u)=v$ forces $u=\tau(\tau(u))=\tau(v)$.

(a) Fix $w\in W_{L}$ and put $y=L^{-1}w$. For $u\in W_{L}$ and $i\in[n]$ let $\lambda_{i}(u)=1$ if $w_{i}<u_{i}$ and $\lambda_{i}(u)=0$ otherwise, let $\lambda(u)\in\mathbb{Z}^{n}$ have components $\lambda_{i}(u)\in\{0,1\}\subseteq\mathbb{Z}$, and let $\tau(u)\in\mathbb{Z}^{n}$ have components $\tau(u)_{i}=w_{i}-u_{i}+L\lambda_{i}(u)$, integers by Arithmetic, Order, Discreteness and Intervals of the Integers §ring as $w_{i},u_{i},L,\lambda_{i}(u)\in\mathbb{Z}$. If $u_{i}\le w_{i}$, then $0\le\tau(u)_{i}=w_{i}-u_{i}\le w_{i}\le L-1$. If $w_{i}<u_{i}$, then $w_{i}+1\le u_{i}$ by Arithmetic, Order, Discreteness and Intervals of the Integers §discrete, that is, $w_{i}-u_{i}\le-1$, and $w_{i}-u_{i}\ge-u_{i}\ge-(L-1)$, so $1\le\tau(u)_{i}\le L-1$. Thus $\tau(u)\in W_{L}$. Moreover $\tau(\tau(u))=u$: if $u_{i}\le w_{i}$, then $\tau(u)_{i}\le w_{i}$, so $\lambda_{i}(\tau(u))=0$ and $\tau(\tau(u))_{i}=w_{i}-(w_{i}-u_{i})=u_{i}$; if $w_{i}<u_{i}$, then $\tau(u)_{i}=w_{i}+(L-u_{i})>w_{i}$ as $u_{i}\le L-1$, so $\lambda_{i}(\tau(u))=1$ and $\tau(\tau(u))_{i}=w_{i}-\tau(u)_{i}+L=u_{i}$. So $\tau$ is a bijection of $W_{L}$, and

$$w-\tau(u)=u+L\bigl(-\lambda(u)\bigr),\qquad L^{-1}\tau(u)=\bigl(y-L^{-1}u\bigr)+\lambda(u)\qquad(u\in W_{L}).$$

Here $w-\tau(u)$ and $-\lambda(u)$ lie in $\mathbb{Z}^{n}$ by Arithmetic, Order, Discreteness and Intervals of the Integers §ring. Hence $\mathfrak{j}(w-\tau(u))=\mathfrak{j}(u)$ by Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols §interaction, applied with $l=-\lambda(u)$. Also $\psi_{k}\in C_{\mathrm{per}}$ by The Trigonometric System on the Torus is Orthonormal §classes, so $\psi_{k}$ is $\mathbb{Z}^{n}$-periodic by Lattice-Periodic Functions and the Periodic Function Classes §classes and Lattice-Periodic Functions and the Periodic Function Classes §periodic, and $\psi_{k}(L^{-1}\tau(u))=\psi_{k}(y-L^{-1}u)$ for every $k\in\mathbb{Z}^{n}$.

(b) Let $p\in[n]$. For $u\in W_{L}$ let $\varrho_{p}u\in\mathbb{Z}^{n}$ have components $(\varrho_{p}u)_{i}=u_{i}$ for $i\ne p$, and $(\varrho_{p}u)_{p}=0$ if $u_{p}=0$, $(\varrho_{p}u)_{p}=L-u_{p}$ if $u_{p}\ne0$; these components are integers by Arithmetic, Order, Discreteness and Intervals of the Integers §ring. If $u_{p}\ne0$ then $0<u_{p}$, as $0\le u_{p}$, so $1\le u_{p}$ by Arithmetic, Order, Discreteness and Intervals of the Integers §discrete; thus $1\le u_{p}\le L-1$, so $1\le L-u_{p}\le L-1$; thus $\varrho_{p}u\in W_{L}$, and $\varrho_{p}(\varrho_{p}u)=u$ since $L-(L-u_{p})=u_{p}$. So $\varrho_{p}$ is a bijection of $W_{L}$. Further $\varrho_{p}u=R_{p}u+L\,\eta(u)$, where $\eta(u)\in\mathbb{Z}^{n}$ has $p$-th component $1$ if $u_{p}\ne0$ and $0$ if $u_{p}=0$, and all other components $0$, so that $\eta(u)\in\mathbb{Z}^{n}$ as $0,1\in\mathbb{Z}$ (Arithmetic, Order, Discreteness and Intervals of the Integers §ring); by Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols §interaction, applied with $l=\eta(u)$ and then with $i=p$, $\mathfrak{j}(\varrho_{p}u)=\mathfrak{j}(R_{p}u)=\mathfrak{j}(u)$. Finally, for $q\in\mathbb{Z}$,

$$S_{q}\bigl(L^{-1}(\varrho_{p}u)_{p}\bigr)=-S_{q}\bigl(L^{-1}u_{p}\bigr).$$

If $u_{p}=0$, both sides are $\pm\sin0=0$ by Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §zero. If $u_{p}\ne0$, put $s=L^{-1}u_{p}$; then $L^{-1}(L-u_{p})=1-s$ and $S_{q}(1-s)=\sin(-2\pi qs+2\pi q)=\sin(-2\pi qs)=-\sin(2\pi qs)=-S_{q}(s)$, by Quarter-Turn Identities and Periodicity of Sine and Cosine §integer and Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §parity.

Step 3: a one-variable expansion. Let $m\in\mathbb{Z}$. By Arithmetic, Order, Discreteness and Intervals of the Integers §sign, exactly one of $m=0$, $0<m$ and $m<0$ holds. By Absolute Value in an Ordered Field, $|m|$ is $m$ or $-m$, so $|m|\in\mathbb{Z}$ by Arithmetic, Order, Discreteness and Intervals of the Integers §ring, and $S_{|m|}$ is defined. Let $\epsilon_{m}=1$ if $0\le m$ and $\epsilon_{m}=-1$ if $m<0$. We show that for all $m\in\mathbb{Z}$ and $t,s\in\mathbb{R}$

$$\phi_{m}(t-s)=\phi_{m}(t)\,C_{m}(s)+\epsilon_{m}\,\phi_{-m}(t)\,S_{|m|}(s).\qquad\text{(A)}$$

If $m=0$: $\phi_{0}(t-s)=1=\phi_{0}(t)\,C_{0}(s)$ and $S_{0}(s)=0$, by The Trigonometric System on the Torus §one-dimensional and Cell Integrals of the Trigonometric Monomials §calculus. If $0<m$: then $|m|=m$ and $-m<0$, so $\phi_{-m}(t)=\sqrt{2}\,S_{m}(t)$; by Addition Formulas for Sine and Cosine §cosine with $a=2\pi mt$ and $x=-2\pi ms$, and by Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §parity,

$$\phi_{m}(t-s)=\sqrt{2}\cos(2\pi mt-2\pi ms)=\sqrt{2}\,C_{m}(t)\,C_{m}(s)+\sqrt{2}\,S_{m}(t)\,S_{m}(s)=\phi_{m}(t)\,C_{m}(s)+\phi_{-m}(t)\,S_{|m|}(s).$$

If $m<0$: put $q=-m=|m|$, so $0<q$, $\phi_{m}(t)=\sqrt{2}\,S_{q}(t)$, $\phi_{-m}(t)=\sqrt{2}\,C_{q}(t)$, and $C_{m}(s)=\cos(-2\pi qs)=C_{q}(s)$ by Values at Zero, Parity, Addition and Double-Angle Identities for Sine and Cosine §parity; by Addition Formulas for Sine and Cosine §sine with $a=2\pi qt$ and $x=-2\pi qs$, and by the same parity clause,

$$\phi_{m}(t-s)=\sqrt{2}\sin(2\pi qt-2\pi qs)=\sqrt{2}\,S_{q}(t)\,C_{q}(s)-\sqrt{2}\,C_{q}(t)\,S_{q}(s)=\phi_{m}(t)\,C_{m}(s)-\phi_{-m}(t)\,S_{|m|}(s).$$

Step 4: eigenvectors (claim 2). Let $k\in\Gamma_{M}$ and $y\in\mathbb{L}_{M}$, let $w\in W_{L}$ with $y=L^{-1}w$, and let $\tau$ be the bijection of Step 2(a) for this $w$. Reindexing along $w'\mapsto L^{-1}w'$ and along $\tau$ (claim 2 of Properties of a Sum over a Finite Index Set), and using Step 2(a),

$$\sum_{z\in\mathbb{L}_{M}}J(y,z)\,\psi_{k}(z)=\sum_{w'\in W_{L}}\mathfrak{j}(w-w')\,\psi_{k}(L^{-1}w')=\sum_{u\in W_{L}}\mathfrak{j}(w-\tau(u))\,\psi_{k}(L^{-1}\tau(u))=\sum_{u\in W_{L}}\mathfrak{j}(u)\,\psi_{k}(y-L^{-1}u).$$

For an integer $p$ with $0\le p\le n$, $i\in[n]$ and $u\in W_{L}$ let $g_{p,i}(u)=\phi_{k_{i}}(y_{i})\,C_{k_{i}}(L^{-1}u_{i})$ if $i\le p$ and $g_{p,i}(u)=\phi_{k_{i}}(y_{i}-L^{-1}u_{i})$ if $p<i$, and let $F_{p}(u)=\prod_{i=1}^{n}g_{p,i}(u)$. As the $i$-th component of $y-L^{-1}u$ is $y_{i}-L^{-1}u_{i}$, $F_{0}(u)=\psi_{k}(y-L^{-1}u)$; and by claim 2 of Properties of Finite Products, $F_{n}(u)=\psi_{k}(y)\prod_{i=1}^{n}C_{k_{i}}(L^{-1}u_{i})$.

Let $p\in[n]$. For $i\ne p$ one has $g_{p-1,i}=g_{p,i}$, since $i\le p-1$ if and only if $i\le p$. Let $r_{i}(u)=g_{p,i}(u)$ for $i\in[n]$ with $i\ne p$, $r_{p}(u)=1$, and $\Pi_{p}(u)=\prod_{i=1}^{n}r_{i}(u)$. Writing each factor $g_{p,i}(u)$, respectively $g_{p-1,i}(u)$, as $r_{i}(u)$ times the number equal to $1$ for $i\ne p$ and to $g_{p,p}(u)$, respectively $g_{p-1,p}(u)$, for $i=p$, claims 2 and 3 of Properties of Finite Products give $F_{p}(u)=\Pi_{p}(u)\,g_{p,p}(u)$ and $F_{p-1}(u)=\Pi_{p}(u)\,g_{p-1,p}(u)$. By (A) with $m=k_{p}$, $t=y_{p}$ and $s=L^{-1}u_{p}$,

$$F_{p-1}(u)=F_{p}(u)+\epsilon_{k_{p}}\,\phi_{-k_{p}}(y_{p})\,D(u),\qquad D(u)=\Pi_{p}(u)\,S_{|k_{p}|}(L^{-1}u_{p}).$$

Each $r_{i}(u)$ depends on $u$ only through $u_{i}$ for $i\ne p$, and $(\varrho_{p}u)_{i}=u_{i}$ for $i\ne p$, so $\Pi_{p}(\varrho_{p}u)=\Pi_{p}(u)$; with Step 2(b) this gives $\mathfrak{j}(\varrho_{p}u)\,D(\varrho_{p}u)=-\mathfrak{j}(u)\,D(u)$. Put $\Sigma=\sum_{u\in W_{L}}\mathfrak{j}(u)\,D(u)$. Reindexing along the bijection $\varrho_{p}$ (claim 2 of Properties of a Sum over a Finite Index Set) and then claim 4 of that lemma give $\Sigma=\sum_{u\in W_{L}}\mathfrak{j}(\varrho_{p}u)\,D(\varrho_{p}u)=\sum_{u\in W_{L}}(-1)\,\mathfrak{j}(u)\,D(u)=-\Sigma$, so $\Sigma=0$. By claims 3 and 4 of Properties of a Sum over a Finite Index Set,

$$\sum_{u\in W_{L}}\mathfrak{j}(u)\,F_{p-1}(u)=\sum_{u\in W_{L}}\mathfrak{j}(u)\,F_{p}(u)+\epsilon_{k_{p}}\,\phi_{-k_{p}}(y_{p})\,\Sigma=\sum_{u\in W_{L}}\mathfrak{j}(u)\,F_{p}(u).$$

By induction on $p$ the sum $\sum_{u\in W_{L}}\mathfrak{j}(u)\,F_{p}(u)$ is the same for all $p$ with $0\le p\le n$. Hence, by claim 4 of Properties of a Sum over a Finite Index Set and Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols §symbol,

$$\sum_{z\in\mathbb{L}_{M}}J(y,z)\,\psi_{k}(z)=\sum_{u\in W_{L}}\mathfrak{j}(u)\,F_{0}(u)=\sum_{u\in W_{L}}\mathfrak{j}(u)\,F_{n}(u)=\psi_{k}(y)\sum_{u\in W_{L}}\mathfrak{j}(u)\prod_{i=1}^{n}C_{k_{i}}(L^{-1}u_{i})=\hat{J}(k)\,\psi_{k}(y).$$

Step 5: the interaction energy (claim 3). Let $f$ and $g$ be lattice fields and $y\in\mathbb{L}_{M}$. By Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §inversion-formula, $g(z)=\sum_{k\in\Gamma_{M}}\hat{g}(k)\,\psi_{k}(z)$ for $z\in\mathbb{L}_{M}$. Using claim 4 of Properties of a Sum over a Finite Index Set, the interchange of claim 5 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, and Step 4,

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

By The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §coefficients, $\sum_{y\in\mathbb{L}_{M}}f(y)\,\psi_{k}(y)=L^{n}\hat{f}(k)$, as $L^{n}L^{-n}=1$. Summing over $y$ and using the same interchange and claim 4 of Properties of a Sum over a Finite Index Set again,

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

Step 6: the bound (claim 4). Let $k\in\Gamma_{M}$ and $u\in W_{L}$. By Cell Integrals of the Trigonometric Monomials §calculus, $|C_{k_{i}}(L^{-1}u_{i})|\le1$ for every $i\in[n]$. By induction on $j\in[n]$, using the recursion of claim 1 of Properties of Finite Products and $|\alpha\beta|=|\alpha|\,|\beta|$ for $\alpha,\beta\in\mathbb{R}$, one has $\bigl|\prod_{i=1}^{j}C_{k_{i}}(L^{-1}u_{i})\bigr|\le1$: for $j=1$ the product is $C_{k_{1}}(L^{-1}u_{1})$, and if the bound holds for $j$ with $j+1\le n$, then the product up to $j+1$ has absolute value $\bigl|\prod_{i=1}^{j}C_{k_{i}}(L^{-1}u_{i})\bigr|\,|C_{k_{j+1}}(L^{-1}u_{j+1})|\le1\cdot1$. Hence $\bigl|\mathfrak{j}(u)\prod_{i=1}^{n}C_{k_{i}}(L^{-1}u_{i})\bigr|\le|\mathfrak{j}(u)|$, and by Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols §symbol, Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §absolute and Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison,

$$|\hat{J}(k)|\le\sum_{u\in W_{L}}\Bigl|\mathfrak{j}(u)\prod_{i=1}^{n}C_{k_{i}}(L^{-1}u_{i})\Bigr|\le\sum_{u\in W_{L}}|\mathfrak{j}(u)|.$$