Lattice and elementary cell (teal) and dual lattice and dual elementary cell (orange)
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Let function $u(\mathbf{x})$, $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ be $2\pi$-periodic with respect to each variable $x_1,x_2,\ldots,x_n$. Then \begin{equation} u(\mathbf{x})= \sum_{\mathbf{m}\in \mathbb{Z}^n} c_{\mathbf{m}} e^{ i \mathbf{m}\cdot \mathbf{x}} \label{eq-4.B.1} \end{equation} with \begin{equation} c_{\mathbf{m}} =(2\pi)^{-n} \iiint_\Omega e^{- i \mathbf{m}\cdot \mathbf{x}} u(\mathbf{x})\,d^nx \label{eq-4.B.2} \end{equation} and \begin{equation} \sum_{\mathbf{m}\in \mathbb{Z}^n} |c_{\mathbf{m}}|^2 =(2\pi)^{-n} \iiint_\Omega |u(\mathbf{x})|^2\,d^n x \label{eq-4.B.3} \end{equation} where $\Omega=(0,2\pi)^n$ is $n$-dimensional cube. Here and below we write $n$-dimensional integral as $\iiint$.
Those formulae follows by repetition of what we have in $1$-dimensional case: we simply make Fourier series decomposition by $x_1$, then by $x_2$ and so on ... ; for example, for $n=2$ \begin{align*} &u(x,y) =\sum_{m\in \mathbb{Z}} u_m(y)e^{ixm}, &&u_m(y)=(2\pi)^{-1}\int_0^{2\pi}u(x,y)e^{-imx}\,dx,\\ &u_m(y)=\sum_{m\in \mathbb{Z}} c_{mn} e^{iny}, && c_{mn}=(2\pi)^{-1}\int_0^{2\pi}u_m(y)e^{-iny}\,dy\\ \implies &u(x,y)=\sum_{(m,n)\in \mathbb{Z}^2}c_{mn} e^{imx+iny}, && c_{mn}=(2\pi)^{-2}\iint_{[0,2\pi]^2} u (x,y) e^{-imx-iny}\,dxdy. \end{align*}
We need slightly generalize these formulae.
Definition 1. Let $\Gamma$ be $n$-dimentional lattice. It means that there are $n$ linearly independent vectors $\mathbf{e}_1, \ldots, \mathbf{e}_n$ and \begin{equation} \Gamma = \{ k_1 \mathbf{e}_1 + k_2 \mathbf{e}_2+\ldots +k_n \mathbf{e}_n:\, k_1,k_2,\ldots,k_n\in \mathbb{Z}\} \label{eq-4.B.4} \end{equation}
Remark 1. The same lattice $\Gamma$ is defined by vectors $\mathbf{e}'_1, \ldots, \mathbf{e}'_n$ with $\mathbf{e}'_j=\sum _k \alpha_{jk}\mathbf{e}_k$ with integer coefficients if and only if the determinant of the matrix $(\alpha_{jk})_{j,k=1,\ldots,n}$ of coefficients is equal $\pm 1$.
Definition 2. Let $\Gamma$ be $n$-dimentional lattice. We call $u(\mathbf{x})$ periodic with respect to $\Gamma$ or simply $\Gamma$-periodic if \begin{equation} u(\mathbf{x}+\mathbf{y})= u(\mathbf{x})\qquad \forall \mathbf{y}\in \Gamma\ \forall \mathbf{x}. \label{eq-4.B.5} \end{equation}
In the previous Subsection $\Gamma= (2\pi\mathbb{Z})^n$. Let us change coordinate system so that $\Gamma$ becomes $(2\pi\mathbb{Z})^n$, apply (\ref{eq-4.B.1})--(\ref{eq-4.B.3}) and then change coordinate system back. We get \begin{equation} u(\mathbf{x})= \sum_{\mathbf{m}\in \Gamma^*} c_{\mathbf{m}} e^{ i \mathbf{m}\cdot \mathbf{x}} \label{eq-4.B.6} \end{equation} with \begin{equation} c_{\mathbf{m}} =|\Omega |^{-1} \iiint_\Omega e^{- i \mathbf{m}\cdot \mathbf{x}} u(\mathbf{x})\,d^n x \label{eq-4.B.7} \end{equation} and \begin{equation} \sum_{\mathbf{m}\in \Gamma^*} |c_{\mathbf{m}}|^2 =|\Omega|^{-1} \iiint_\Omega |u(\mathbf{x})|^2\,d^n x \label{eq-4.B.8} \end{equation} where $|\Omega|$ is a volume of $\Omega$ and
Definition 3.
Remark 2. We prefer to use original coordinate system rather than one with coordinate vectors $(2\pi)^{-1}\mathbf{e}_1,\ldots, (2\pi)^{-1}\mathbf{e}_n$ because the latter is not necessarily orthonormal and in it Laplacian will have a different form.
These notions are important for studying the band spectrum of the Schrödinger operator $-\Delta +V(\mathbf{x})$ with periodic (with respect to some lattice $\Gamma$) potential in the whole space which has applications to the Physics of crystals. For this the following decomposition is used for functions $u(\mathbf{x})$ in the whole space $\mathbb{R}^n$.
Theorem 1. Let $u(\mathbf{x})$ be sufficiently fast decaying function on $\mathbb{R}^n$. Then \begin{equation} u(\mathbf{x})= \iiint_{\Omega^*} u(\mathbf{k};\mathbf{x})\,d^n\mathbf{k} \label{eq-4.B.10} \end{equation} with \begin{equation} u(\mathbf{k};\mathbf{x})= (2\pi)^{-n}|\Omega| \sum_{\mathbf{l}\in \Gamma} e^{-i\mathbf{k}\cdot \mathbf{l}} u(\mathbf{x}+\mathbf{l}). \label{eq-4.B.11} \end{equation} Here $u(\mathbf{k};\mathbf{x})$ is quasiperiodic with quasimomentum $\mathbf{k}$ \begin{equation} u(\mathbf{k};\mathbf{x}+\mathbf{y})= e^{i\mathbf{k}\cdot\mathbf{y}}u(\mathbf{k};\mathbf{x})\qquad \forall \mathbf{y}\in \Gamma\ \forall \mathbf{x}. \label{eq-4.B.12} \end{equation}
Lattice and elementary cell (teal) and dual lattice and dual elementary cell (orange)
Proof. Observe that since $u$ is sufficiently fast decaying series in (\ref{eq-4.B.11}) converges and one can see easily that it defines quasiperiodic with quasimomentum $\mathbf{k}$ function.
The proof of (\ref{eq-4.B.10}) is trivial because \begin{gather*} \iiint_{\Omega^*} e^{-i\mathbf{k}\cdot \mathbf{l}} \,d^n\mathbf{k}= \left\{\begin{aligned} &|\Omega^*| &&\mathbf{l}=0,\\ &0 &&0\ne \mathbf{l}\in \Gamma \end{aligned}\right. \end{gather*} and $|\Omega^*|=(2\pi)^n|\Omega|^{-1}$. and $|\Omega^*|=(2\pi)^n|\Omega|^{-1}$.
Remark 3. Since choice of vectors $\mathbf{e}_1, \ldots, \mathbf{e}_n$ for given lattice $\Gamma$ is not unique, the choice of $\Omega$ and $\Omega^*$ is not unique either, but $|\Omega|$ is defined uniquely.