Appendix B. Fourier transforms, Fourier series, and sets of wavevectors

https://doi.org/10.48550/arXiv.2403.13981

This appendix outlines some notation and normalization conventions that are used for Fourier transforms and Fourier series.

One simple convention that is used throughout this work is that if $f$ is a function of space (i.e., one or more positions) and time, then its Fourier transforms in space, time, and both space and time are denoted by ${\ftsf}$, ${\ftt{f}}$, and ${\ftst{f}}$, respectively.

If $\domain$ is the spatial domain of each component of the position argument of $f$, the domain of each wavevector component of the argument of ${\ftsf}$ or ${\ftst{f}}$ will be denoted by ${\ftsdomain}$.

It will be assumed that the time domain of $f$ is ${\realone}$; and the frequency domain of ${\ftt{f}}$ or ${\ftst{f}}$ is also ${\realone}$ (or ${\realnonneg}$ if frequencies are required to be positive) but will be denoted by ${\ftt{\realone}}$.

To reduce clutter in expressions for unitary Fourier transforms, let ${\displaystyle \fourierconst\equiv 1/\sqrt{2\pi}}$.

B.1 Fourier transforms with respect to position, time, and both position and time

Let ${\domain\in\{\realone,\onetorus\}}$; let ${\image\in\{\realone,\complex\}}$; and let
\begin{align*} f:\domain^m\times \realone\to \image; (u,t)\mapsto f(u,t), \end{align*}
where each value of ${u\in\domain^m}$ specifies one or more positions and ${t\in\realone}$ is a time variable.

B.1.1 Fourier transform with respect to position

The spatial Fourier transform exists if the function ${f(t):\domain^m\to\image; u\mapsto f(u,t)}$ is an element of ${\lebesgone(\domain^m,\image)}$ at all values of ${t\in\realone}$, i.e., if
\begin{align*} \int_{\domain^m}\dd{u} \abs{f(u,t)} < \infty,\;\forall t\in\realone. \end{align*}

The unitary Fourier transform with respect to space is effected by the operator,

\begin{align*} &\fouriernoarg_s:\lebesgone(\domain^m\times\realone)\to\lebesgone(\ftsdomain^m\times\realone); f\mapsto \ftsf\equiv \fourierspace{f}, \end{align*}
where ${\ftsdomain=\realone}$ if ${\domain=\realone}$, and ${\ftsdomain= (2\pi/C)\integer}$ if ${\domain=\onetorus(C)}$.

The unitary Fourier transform of $f$ with respect to position is the function

\begin{align*} &\fourierspace{f}\equiv\ftsf: \ftsdomain^m\times\realone\to\image, \\ &(q,t)\mapsto \ftsf(q,t)\equiv \fourierconst^m\int_{\domain^m} \dd{u} f(u,t)e^{-i q\cdot u}, \tag{104} \end{align*}
where ${q\in\ftsdomain^m\cong \domain^m}$; and the inverse of Eq. (104) is
\begin{align*} f(u,t) \equiv \fourierconst^m\int_{\domain^m} \dd{q} \fts{f}(q,t)e^{i q\cdot u}. \end{align*}

B.1.2 Fourier transform with respect to time

The Fourier transform of $f$ with respect to time exists if the function ${f(u):\realone\to\image; t\mapsto f(u,t)}$ is an element of ${\lebesgone(\realone,\image)}$ at all values of ${u\in\domain^m}$, i.e., if
\begin{align*} \int_{\realone}\dd{t} \abs{f(u,t)} < \infty,\;\forall u\in\domain^m. \end{align*}

The unitary Fourier transform with respect to time is effected by the operator

\begin{align*} &\fouriernoarg_t:\lebesgone(\domain^m\times\realone)\to\lebesgone(\domain^m\times\ftt{\realone}); \; f\mapsto \ftt{f}\equiv \fouriertime{f}, \end{align*}
where ${\ftt{\realone}\cong\realone}$.

The unitary Fourier transform of $f$ with respect to time is the function

\begin{align*} &\fouriertime{f}\equiv\ftt{f}: \domain^m\times\ftt{\realone}\to\image, \\ &(u,\omega)\mapsto \ftt{f}(u,\omega)\equiv \fourierconst\int_{\realone} \dd{t} f(u,t)e^{i \omega t}, \tag{105} \end{align*}
where ${\omega\in\ftt{\realone}\cong \realone}$. The inverse of Eq. (105) is
\begin{align*} f(u,t) \equiv \fourierconst\int_{\ftt{\realone}} \dd{\omega} \ftt{f}(u,\omega)e^{-i \omega t}. \end{align*}

B.1.3 Fourier transform with respect to position and time

The Fourier transform of $f$ with respect to position and time exists if ${f\in\lebesgone(\domain^m\times\realone,\image)}$, i.e., if
\begin{align*} \int_{\domain^m}\dd{u} \int_{\realone}\dd{t} \abs{f(u,t)} < \infty. \end{align*}

The unitary Fourier transform with respect to position and time is effected by the operator

\begin{align*} &\fouriernoarg_{st}:\lebesgone(\domain^m\times\realone)\to\lebesgone(\ftsdomain^m\times\ftt{\realone}); \; f\mapsto \ftst{f}\equiv \fourierspacetime{f}, \end{align*}
where ${\ftt{\realone}\cong\realone}$; and ${\ftsdomain=\realone}$ if ${\domain=\realone}$, and ${\ftsdomain= (2\pi/C)\integer}$ if ${\domain=\onetorus(C)}$.

The unitary Fourier transform of $f$ with respect to position and time is the function

\begin{align*} &\fourierspacetime{f}\equiv\ftst{f}: \ftsdomain^m\times\ftt{\realone}\to\image, \\ &(q,\omega)\mapsto \ftst{f}(q,\omega)\equiv \fourierconst^{m+1}\int_{\domain^m}\dd{u}\int_{\realone} \dd{t} f(u,t) e^{-i (q\cdot u-\omega t)}, \end{align*}
where ${\omega\in\ftt{\realone}}$; and the expression for the inverse transform is
\begin{align*} f(u,t) \equiv \fourierconst^{m+1}\int_{\ftsdomain^m}\dd{q}\int_{\ftt{\realone}} \dd{\omega} \ftst{f}(q,\omega)e^{i (q\cdot u-\omega t)}. \end{align*}

B.2 Wavevectors compatible with $\volume$ and $\bulksize$ periodicities

Definitions of the wavevector differentials, ${\hbulksize}$ and ${\hreciplatt}$, and the sets of wavevectors, ${\reciplattg}$, $\BZ$, and ${\reciplatt}$, follow. Although all five are determined by the values of one or both of ${\bulksize}$ and $\volume$, these dependences will not usually be made explicit. For example, to make mathematical expressions less cluttered, ${\hbulksize(\bulksize)}$ will usually be denoted as ${\hbulksize}$.

It will sometimes be important to be conscious of the implicit dependences of ${\hbulksize}$, ${\hreciplatt}$, ${\reciplattg}$, $\BZ$, and ${\reciplatt}$ on ${\bulksize}$ and/or $\volume$.

The smallest wavevectors that are compatible with $\bulksize$-periodicity have magnitude

\begin{align*} \hbulksize=\hbulksize(\bulksize)\equiv 2\pi/\bulksize, \end{align*}
and the set of all wavevectors that are compatible with ${\bulksize}$-periodicity is
\begin{align*} \reciplattg=\reciplattg(\bulksize)\equiv \hbulksize\integer\equiv\left\{g=m_g\hbulksize:m_g\in\integer\right\}. \end{align*}

The subset of ${\reciplattg}$ that comprises all wavevectors in the first Brillouin zone that are compatible with $\bulksize$-periodicity is

\begin{align*} \BZ=\BZ(\bulksize,\volume)\equiv \left\{ k \in \reciplattg : -\frac{\pi}{\volume} < k \leq \frac{\pi}{\volume}\right\}. \end{align*}

The magnitude of the smallest wavevectors that are compatible with $\volume$-periodicity is

\begin{align*} \hreciplatt=\hreciplatt(\volume)\equiv 2\pi/\volume; \end{align*}
and the reciprocal lattice,
\begin{align*} \reciplatt=\reciplatt(\volume)\equiv \hreciplatt\integer, \end{align*}
is the set of all wavevectors that are compatible with ${\volume}$-periodicity.

Every element of ${\reciplattg}$ can be expressed as the sum of a reciprocal lattice vector and an element of $\BZ$, i.e.,

\begin{align*} \reciplattg = \left\{g=G+k : G\in\reciplatt,\; k \in\BZ\right\}. \end{align*}

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