On the Separation of Variables for the Modular XXZ Magnet and the Lattice Sinh-Gordon Models

Abstract

We construct the generalised eigenfunctions of the entries of the monodromy matrix of the N-site modular XXZ magnet and show, in each case, that these form a complete orthogonal system in $L^2(\mathbb {R}^N)$. In particular, we develop a new and simple technique, allowing one to prove the completeness of such systems. As a corollary of our analysis, we prove the Bytsko–Teschner conjecture relative to the structure of the spectrum of the $\varvec{ \texttt {B} } (\lambda )$-operator for the odd length lattice Sinh-Gordon model.

Appendices

Appendix A. Main Notations

• N-dimensional vectors are denoted as $\varvec{x}_N=(x_1,\dots , x_N)$;

• $N-1$ dimensional vectors built from an N-dimensional vector $\varvec{x}_N$ with the removed $m^{\mathrm {th}}$ coordinate are denoted as $\varvec{x}_{N;[m]}$ and read

  $$\begin{aligned} \varvec{x}_{N;[m]} \, = \, \big ( x_1,\dots , x_{m-1},x_{m+1},\dots , x_N \big ) \;. \end{aligned}$$

  (238)

• Given an N-dimensional vector $\varvec{x}_N$, we denote

  $$\begin{aligned} \overline{\varvec{x}}_N=\sum \limits _{a=1}^{N}x_a \; . \end{aligned}$$

  (239)

• Ratios of products of one variable functions appearing with multi-component entries are denoted using the hypergeometric notations, e.g.

  $$\begin{aligned} f\left( \begin{array}{cc} a_1,\dots , a_n \\ b_1,\dots , b_m \end{array}\right) \, = \, { \mathchoice{\dfrac{ \prod \nolimits _{k=1}^{n} f(a_k) }{\prod \nolimits _{k=1}^{m} f(b_k) }}{\dfrac{ \prod \nolimits _{k=1}^{n} f(a_k) }{\prod \nolimits _{k=1}^{m} f(b_k) }}{\frac{ \prod \nolimits _{k=1}^{n} f(a_k) }{\prod \nolimits _{k=1}^{m} f(b_k) }}{\frac{ \prod \nolimits _{k=1}^{n} f(a_k) }{\prod \nolimits _{k=1}^{m} f(b_k) }} } \;. \end{aligned}$$

  (240)

• Given indexed symbols $x_{a},x_{b}$, we denote $x_{ab}=x_a-x_b$.

• Given $y \in \mathbb {C}$, $y^*$ stands for its complex conjugate and $y^{\star }=-y-{\mathrm i}\tfrac{\Omega }{2}$.

Appendix B. Properties of the Auxiliary Special Functions

1.1 B.1. The Quantum Dilogarithm and the $D_{\alpha }$ Functions

The quantum dilogarithm $\varpi$ is a meromorphic function on $\mathbb {C}$ which admits the integral representation

$$\begin{aligned} \varpi (\lambda ) \; = \; \exp \Bigg \{ \pm { \mathchoice{\dfrac{ {\mathrm i}\pi }{ 2 \omega _1 \omega _2 }}{\dfrac{ {\mathrm i}\pi }{ 2 \omega _1 \omega _2 }}{\frac{ {\mathrm i}\pi }{ 2 \omega _1 \omega _2 }}{\frac{ {\mathrm i}\pi }{ 2 \omega _1 \omega _2 }} } \cdot \Big ( \lambda ^2 \,+\, { \mathchoice{\dfrac{ \omega ^2_1+\omega ^2_2 }{ 12 }}{\dfrac{ \omega ^2_1+\omega ^2_2 }{ 12 }}{\frac{ \omega ^2_1+\omega ^2_2 }{ 12 }}{\frac{ \omega ^2_1+\omega ^2_2 }{ 12 }} } \Big ) \; - \; {\mathrm i}\int \limits _{ \mathbb {R}\pm {\mathrm i}0^+}^{} { \mathchoice{\dfrac{ \mathrm {d}t }{ 4 t }}{\dfrac{ \mathrm {d}t }{ 4 t }}{\frac{ \mathrm {d}t }{ 4 t }}{\frac{ \mathrm {d}t }{ 4 t }} } { \mathchoice{\dfrac{ \mathrm {e}^{-2{\mathrm i}\lambda t} }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }}{\dfrac{ \mathrm {e}^{-2{\mathrm i}\lambda t} }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }}{\frac{ \mathrm {e}^{-2{\mathrm i}\lambda t} }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }}{\frac{ \mathrm {e}^{-2{\mathrm i}\lambda t} }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }} } \Bigg \} \;, \end{aligned}$$

(241)

valid for $|\mathfrak {I}(\lambda )| \, < \, \Omega /2$.

This function is self-dual and satisfies the first-order finite-difference equations

$$\begin{aligned} \varpi \big (\lambda +{\mathrm i}\omega _2\big )= & {} 2{\mathrm i}\sinh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\tau }{2}}{\dfrac{\tau }{2}}{\frac{\tau }{2}}{\frac{\tau }{2}} } \big ) \Big ] \cdot \varpi (\lambda )\nonumber \\&\quad \mathrm {and} \quad \varpi \big (\lambda +{\mathrm i}\omega _1\big ) \, = \, 2{\mathrm i}\sinh \Big [ { \mathchoice{\dfrac{\pi }{\omega _2}}{\dfrac{\pi }{\omega _2}}{\frac{\pi }{\omega _2}}{\frac{\pi }{\omega _2}} }\big (\lambda - {\mathrm i}{ \mathchoice{\dfrac{\tau }{2}}{\dfrac{\tau }{2}}{\frac{\tau }{2}}{\frac{\tau }{2}} } \big ) \Big ] \cdot \varpi (\lambda ) \;. \nonumber \\ \end{aligned}$$

(242)

From there, one entails that

$$\begin{aligned}&\varpi \Big (\lambda -{\mathrm i}\tfrac{ \Omega }{ 2 } + {\mathrm i}\ell \omega _1 + {\mathrm i}k \omega _2 \Big ) = (-1)^{k\ell } \big ( -2{\mathrm i}\big )^{\ell + k} \cdot \prod \limits _{p=0}^{k-1} \sinh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda + {\mathrm i}p \omega _2 \big ) \Big ] \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \times \prod \limits _{p=0}^{\ell -1} \sinh \Big [ { \mathchoice{\dfrac{\pi }{\omega _2}}{\dfrac{\pi }{\omega _2}}{\frac{\pi }{\omega _2}}{\frac{\pi }{\omega _2}} }\big (\lambda + {\mathrm i}p \omega _1 \big ) \Big ] \cdot \varpi \Big (\lambda -{\mathrm i}\tfrac{ \Omega }{ 2 } \Big ) \nonumber \\ \end{aligned}$$

(243)

and symmetrically,

$$\begin{aligned}&\varpi \Big (\lambda -{\mathrm i}\tfrac{ \Omega }{ 2 } - {\mathrm i}\ell \omega _1 - {\mathrm i}k \omega _2 \Big ) = (-1)^{k\ell } \Big ( { \mathchoice{\dfrac{{\mathrm i}}{2}}{\dfrac{{\mathrm i}}{2}}{\frac{{\mathrm i}}{2}}{\frac{{\mathrm i}}{2}} } \Big )^{\ell + k} \cdot \prod \limits _{p=1}^{k} \Big \{ \sinh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda - {\mathrm i}p \omega _2 \big ) \Big ] \Big \}^{-1} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \times \prod \limits _{p=1}^{\ell } \Big \{ \sinh \Big [ { \mathchoice{\dfrac{\pi }{\omega _2}}{\dfrac{\pi }{\omega _2}}{\frac{\pi }{\omega _2}}{\frac{\pi }{\omega _2}} }\big (\lambda - {\mathrm i}p \omega _1 \big ) \Big ] \Big \}^{-1} \cdot \varpi \Big (\lambda -{\mathrm i}\tfrac{ \Omega }{ 2 } \Big ) \;. \nonumber \\ \end{aligned}$$

(244)

The quantum dilogarithm has only simple poles and zeroes. These are located at

$$\begin{aligned}&\varpi (x)=0 \quad \mathrm {iff} \quad x \in {\mathrm i}{ \mathchoice{\dfrac{\Omega }{2}}{\dfrac{\Omega }{2}}{\frac{\Omega }{2}}{\frac{\Omega }{2}} } + {\mathrm i}\mathbb {N} \omega _1+ {\mathrm i}\mathbb {N} \omega _2\nonumber \\&\qquad \mathrm {and} \varpi ^{-1}(x)=0 \quad \mathrm {iff} \quad x \in -{\mathrm i}{ \mathchoice{\dfrac{\Omega }{2}}{\dfrac{\Omega }{2}}{\frac{\Omega }{2}}{\frac{\Omega }{2}} } - {\mathrm i}\mathbb {N} \omega _1 - {\mathrm i}\mathbb {N} \omega _2. \end{aligned}$$

(245)

$\varpi$ satisfies the inversion identity $\varpi (\lambda )\varpi (-\lambda )=1$ and $\big (\varpi (\lambda ^*)\big )^*=\varpi ^{-1}(\lambda )$. One can also establish that

$$\begin{aligned} \mathrm {Res}\Big (\varpi \big ( \lambda - {\mathrm i}\tfrac{\Omega }{2}\big ) \cdot \mathrm {d}\lambda , \lambda =0 \Big ) \, = \, { \mathchoice{\dfrac{{\mathrm i}}{2\pi }}{\dfrac{{\mathrm i}}{2\pi }}{\frac{{\mathrm i}}{2\pi }}{\frac{{\mathrm i}}{2\pi }} } \sqrt{\omega _1 \omega _2} \qquad \mathrm {and} \qquad \varpi ( \tfrac{{\mathrm i}}{2} \tau ) \; = \; \sqrt{ { \mathchoice{\dfrac{\omega _2}{\omega _1 }}{\dfrac{\omega _2}{\omega _1 }}{\frac{\omega _2}{\omega _1 }}{\frac{\omega _2}{\omega _1 }} } } \;. \end{aligned}$$

(246)

The above entails that, for $(k,\ell )\in \mathbb {N}^2$,

$$\begin{aligned}&\mathrm {Res}\Big (\varpi \big ( \lambda - {\mathrm i}\tfrac{\Omega }{2}\big ) \cdot \mathrm {d}\lambda , \lambda \nonumber \\&\quad =-{\mathrm i}\ell \omega _1 -{\mathrm i}k \omega _2 \Big ) \, = \, { \mathchoice{\dfrac{{\mathrm i}}{2 \pi }}{\dfrac{{\mathrm i}}{2 \pi }}{\frac{{\mathrm i}}{2 \pi }}{\frac{{\mathrm i}}{2 \pi }} } \sqrt{\omega _1 \omega _2} (-1)^{k\ell } \Big ( { \mathchoice{\dfrac{ 1 }{2 {\mathrm i}}}{\dfrac{ 1 }{2 {\mathrm i}}}{\frac{ 1 }{2 {\mathrm i}}}{\frac{ 1 }{2 {\mathrm i}}} } \Big )^{\ell + k} \nonumber \\&\qquad \times \bigg \{ \prod \limits _{p=1}^{k} \sinh \Big [ {\mathrm i}p \pi { \mathchoice{\dfrac{ \omega _2 }{\omega _1}}{\dfrac{ \omega _2 }{\omega _1}}{\frac{ \omega _2 }{\omega _1}}{\frac{ \omega _2 }{\omega _1}} } \Big ] \cdot \prod \limits _{p=1}^{\ell } \sinh \Big [ {\mathrm i}p \pi { \mathchoice{\dfrac{\omega _1 }{\omega _2}}{\dfrac{\omega _1 }{\omega _2}}{\frac{\omega _1 }{\omega _2}}{\frac{\omega _1 }{\omega _2}} } \Big ] \bigg \}^{-1} \;. \end{aligned}$$

(247)

The function $D_{\alpha }$ is defined by the below ratio of dilogarithms

$$\begin{aligned} D_{\alpha }(\lambda ) \; = \; { \mathchoice{\dfrac{\varpi (\lambda +\alpha ) }{ \varpi (\lambda -\alpha ) }}{\dfrac{\varpi (\lambda +\alpha ) }{ \varpi (\lambda -\alpha ) }}{\frac{\varpi (\lambda +\alpha ) }{ \varpi (\lambda -\alpha ) }}{\frac{\varpi (\lambda +\alpha ) }{ \varpi (\lambda -\alpha ) }} } \qquad \mathrm {so}\; \mathrm {that} \qquad D_{\alpha }(\lambda ) \, = \, \Big ( D_{-\alpha ^*}(\lambda ^*) \Big )^* \;. \end{aligned}$$

(248)

The function $D_{\alpha }$ is a meromorphic function on $\mathbb {C}$ that admits a holomorphic determination of the logarithm on

$$\begin{aligned} \mathcal {S}_{ { \mathchoice{\dfrac{\Omega }{2}}{\dfrac{\Omega }{2}}{\frac{\Omega }{2}}{\frac{\Omega }{2}} }-|\mathfrak {I}(\alpha )| }(\mathbb {R}) \; = \; \Big \{ \lambda \in \mathbb {C}\, : \, \big | \mathfrak {I}(\lambda ) \big | \; < \; { \mathchoice{\dfrac{1}{2}}{\dfrac{1}{2}}{\frac{1}{2}}{\frac{1}{2}} } \Big ( \omega _1 \, + \, \omega _2 \Big ) - \, \big | \mathfrak {I}(\alpha ) \big | \Big \} \end{aligned}$$

(249)

given by

$$\begin{aligned} \ln D_{\alpha }(\lambda ) \; = \; \mp { \mathchoice{\dfrac{ 2 {\mathrm i}\pi }{ \omega _1 \omega _2 }}{\dfrac{ 2 {\mathrm i}\pi }{ \omega _1 \omega _2 }}{\frac{ 2 {\mathrm i}\pi }{ \omega _1 \omega _2 }}{\frac{ 2 {\mathrm i}\pi }{ \omega _1 \omega _2 }} } \alpha \lambda \; + \; {\mathrm i}\int \limits _{ \mathbb {R}\pm {\mathrm i}0^+}^{} { \mathchoice{\dfrac{ \mathrm {d}t }{ 2 t }}{\dfrac{ \mathrm {d}t }{ 2 t }}{\frac{ \mathrm {d}t }{ 2 t }}{\frac{ \mathrm {d}t }{ 2 t }} } { \mathchoice{\dfrac{ \mathrm {e}^{2{\mathrm i}\lambda t} \sin (2\alpha t ) }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }}{\dfrac{ \mathrm {e}^{2{\mathrm i}\lambda t} \sin (2\alpha t ) }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }}{\frac{ \mathrm {e}^{2{\mathrm i}\lambda t} \sin (2\alpha t ) }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }}{\frac{ \mathrm {e}^{2{\mathrm i}\lambda t} \sin (2\alpha t ) }{ \sinh \big (t\omega _1 \big ) \cdot \sinh \big (t\omega _2 \big ) }} } \;. \end{aligned}$$

(250)

The function $D_{\alpha }(\lambda )$ satisfies the properties

• $D_{\alpha }(\lambda )$ is self-dual, namely invariant under the exchange $\omega _1\leftrightarrow \omega _2$;

• for $\omega _2>\omega _1$ it admits the asymptotic behaviours

  $$\begin{aligned} D_{\alpha }(\lambda )= & {} \mathrm {e}^{ \mp { \mathchoice{\dfrac{ 2 {\mathrm i}\pi }{ \omega _1\omega _2}}{\dfrac{ 2 {\mathrm i}\pi }{ \omega _1\omega _2}}{\frac{ 2 {\mathrm i}\pi }{ \omega _1\omega _2}}{\frac{ 2 {\mathrm i}\pi }{ \omega _1\omega _2}} } \cdot \lambda \alpha }\cdot \left( 1 \; + \; \mathrm {O}\Big ( \mathrm {e}^{ \mp { \mathchoice{\dfrac{ 2\pi }{ \omega _2 }}{\dfrac{ 2\pi }{ \omega _2 }}{\frac{ 2\pi }{ \omega _2 }}{\frac{ 2\pi }{ \omega _2 }} } \lambda } \sinh \Big [ { \mathchoice{\dfrac{2\pi }{ \omega _2}}{\dfrac{2\pi }{ \omega _2}}{\frac{2\pi }{ \omega _2}}{\frac{2\pi }{ \omega _2}} } \alpha \Big ] \Big ) \right) \nonumber \\&\quad \mathrm {when} \quad \lambda \rightarrow \infty \; , \; \; |\arg (\pm \lambda )| \, < \, { \mathchoice{\dfrac{\pi }{2}}{\dfrac{\pi }{2}}{\frac{\pi }{2}}{\frac{\pi }{2}} } ; \end{aligned}$$

  (251)

• $D_{\alpha }(\lambda )$ satisfies the difference equation

  $$\begin{aligned} { \mathchoice{\dfrac{D_{\alpha }(\lambda +{\mathrm i}\omega _2)}{D_{\alpha }(\lambda ) }}{\dfrac{D_{\alpha }(\lambda +{\mathrm i}\omega _2)}{D_{\alpha }(\lambda ) }}{\frac{D_{\alpha }(\lambda +{\mathrm i}\omega _2)}{D_{\alpha }(\lambda ) }}{\frac{D_{\alpha }(\lambda +{\mathrm i}\omega _2)}{D_{\alpha }(\lambda ) }} } \; = \; { \mathchoice{\dfrac{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }+\alpha \big ) \Big ] }{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }-\alpha \big ) \Big ] }}{\dfrac{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }+\alpha \big ) \Big ] }{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }-\alpha \big ) \Big ] }}{\frac{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }+\alpha \big ) \Big ] }{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }-\alpha \big ) \Big ] }}{\frac{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }+\alpha \big ) \Big ] }{ \cosh \Big [ { \mathchoice{\dfrac{\pi }{\omega _1}}{\dfrac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}}{\frac{\pi }{\omega _1}} }\big (\lambda +{\mathrm i}{ \mathchoice{\dfrac{\omega _2}{2}}{\dfrac{\omega _2}{2}}{\frac{\omega _2}{2}}{\frac{\omega _2}{2}} }-\alpha \big ) \Big ] }} } \end{aligned}$$

  (252)

  as well as its dual $\omega _1\leftrightarrow \omega _2$;

• $D_{\alpha }(\lambda )$ enjoys the transmutation properties

  $$\begin{aligned} D_{\alpha }\Big ( \lambda \mp {\mathrm i}\tfrac{\omega _2}{2} \Big ) \, = \, { \mathchoice{\dfrac{ D_{\alpha - {\mathrm i}\frac{\omega _2}{2} }\big ( \lambda \big ) }{ 2 \cosh \Big [\frac{\pi }{\omega _1}(\lambda \pm \alpha ) \Big ] }}{\dfrac{ D_{\alpha - {\mathrm i}\frac{\omega _2}{2} }\big ( \lambda \big ) }{ 2 \cosh \Big [\frac{\pi }{\omega _1}(\lambda \pm \alpha ) \Big ] }}{\frac{ D_{\alpha - {\mathrm i}\frac{\omega _2}{2} }\big ( \lambda \big ) }{ 2 \cosh \Big [\frac{\pi }{\omega _1}(\lambda \pm \alpha ) \Big ] }}{\frac{ D_{\alpha - {\mathrm i}\frac{\omega _2}{2} }\big ( \lambda \big ) }{ 2 \cosh \Big [\frac{\pi }{\omega _1}(\lambda \pm \alpha ) \Big ] }} } \;; \end{aligned}$$

  (253)

• $D_{\alpha }(\lambda )$ has simple zeroes at

  $$\begin{aligned} \pm \Big \{ -\alpha \, + \, {\mathrm i}{ \mathchoice{\dfrac{\Omega }{2}}{\dfrac{\Omega }{2}}{\frac{\Omega }{2}}{\frac{\Omega }{2}} }\, + \, {\mathrm i}n \omega _1 \, + \, {\mathrm i}m \omega _2\; : \; (n,m) \in \mathbb {N}^2 \Big \} \; ; \end{aligned}$$

  (254)

• $D_{\alpha }(\lambda )$ has simple poles at

  $$\begin{aligned} \pm \Big \{ \alpha \, + \, {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{2}}{\dfrac{ \Omega }{2}}{\frac{ \Omega }{2}}{\frac{ \Omega }{2}} } \, + \, {\mathrm i}n \omega _1 \, + \, {\mathrm i}m \omega _2\; : \; (n,m) \in \mathbb {N}^2 \Big \} \;. \end{aligned}$$

  (255)

1.2 B.2. Integral Identities

The function $D_{\alpha }$ admits the Fourier transform

$$\begin{aligned} \mathcal {F}[D_{\alpha }](t) \; = \; \sqrt{\omega _1\omega _2} \cdot \mathcal {A}(\alpha ) \cdot D_{\alpha ^{\star }}\Big ( { \mathchoice{\dfrac{\omega _1\omega _2}{2\pi }}{\dfrac{\omega _1\omega _2}{2\pi }}{\frac{\omega _1\omega _2}{2\pi }}{\frac{\omega _1\omega _2}{2\pi }} } t \Big ) \end{aligned}$$

(256)

with

$$\begin{aligned} \mathcal {F}[f](t)\;= \; \int \limits _{ \mathbb {R}}^{} \mathrm {e}^{it x} f(x) \cdot \mathrm {d}x \qquad \mathrm {for} \quad f \in L^{1}(\mathbb {R},\mathrm {d}x) \;. \end{aligned}$$

(257)

Here, we remind that

$$\begin{aligned} \mathcal {A}(\alpha ) \; = \; \varpi \left( 2 \alpha + {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{2}}{\dfrac{ \Omega }{2}}{\frac{ \Omega }{2}}{\frac{ \Omega }{2}} } \right) \qquad \mathrm {and} \qquad \alpha ^{\star }=-\alpha -{\mathrm i}{ \mathchoice{\dfrac{ \Omega }{2}}{\dfrac{ \Omega }{2}}{\frac{ \Omega }{2}}{\frac{ \Omega }{2}} } \;. \end{aligned}$$

(258)

(256) entails that

$$\begin{aligned} \lim _{\alpha \rightarrow 0} \Big \{ \mathcal {A}(\alpha ) D_{\alpha ^{\star }}(t) \Big \} \, = \, \delta (t) \;. \end{aligned}$$

(259)

The $D_{\alpha }$ functions satisfy the three-term integral identity [16]

$$\begin{aligned}&\int \limits _{ \mathbb {R}}^{} D_{\alpha }\big ( \omega _1 \omega _2 (x-u) \big ) \cdot D_{\beta }\big ( \omega _1 \omega _2 (x-v)\big ) \cdot D_{\gamma }\big ( \omega _1 \omega _2 (x-w) \big )\cdot \mathrm {d}x \nonumber \\&\quad = { \mathchoice{\dfrac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }}{\dfrac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }}{\frac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }}{\frac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }} } D_{\alpha ^{\star }}\big ( \omega _1 \omega _2(w-v)\big )\cdot D_{\beta ^{\star }}\big ( \omega _1 \omega _2(u-w)\big ) \nonumber \\&\qquad \times D_{\gamma ^{\star }}\big ( \omega _1 \omega _2(u-v) \big ) \end{aligned}$$

(260)

provided that $\alpha +\beta +\gamma =-{\mathrm i}\Omega$ and [5]

$$\begin{aligned}&\int \limits _{ \mathbb {R}}^{} D_{\alpha }\big ( \omega _1 \omega _2 (x-u)\big )\cdot D_{\beta }\big ( \omega _1 \omega _2 (x-v)\big )\cdot D_{\gamma }\big ( \omega _1 \omega _2 (x-w)\big )\cdot D_{\delta }\big ( \omega _1 \omega _2 (x-z)\big )\cdot \mathrm {d}x \nonumber \\&\quad = \mathcal {A}\big (\alpha ,\beta ,\gamma ,\delta \big ) \cdot D_{\alpha + \beta + {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{ 2 }}{\dfrac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }} } } \bigg ( \begin{array}{c} \omega _1 \omega _2(u - v) \\ \omega _1 \omega _2 (w - z) \end{array}\bigg ) \nonumber \\&\qquad \times \int \limits _{ \mathbb {R}}^{} D_{\alpha ^{\star }}\big ( \omega _1 \omega _2 (x-v)\big )\cdot D_{\beta ^{\star }}\big ( \omega _1 \omega _2 (x-u) \big )\cdot D_{\gamma ^{\star }}\big ( \omega _1 \omega _2 (x-z)\big )\nonumber \\&\qquad \times D_{\delta ^{\star }}\big ( \omega _1 \omega _2 (x-w)\big )\cdot \mathrm {d}x \end{aligned}$$

(261)

provided that $\alpha +\beta +\gamma +\delta \, = \, -{\mathrm i}\Omega$.

Sending one of the integration variables to infinity provides one with the auxiliary identities

$$\begin{aligned}&\int \limits _{ \mathbb {R}}^{} D_{\alpha }\big ( \omega _1 \omega _2(x-u) \big ) \cdot \mathrm {e}^{ \pm 2 {\mathrm i}\pi \beta x} \cdot D_{\gamma }\big ( \omega _1 \omega _2 (x-w) \big )\cdot \mathrm {d}x \nonumber \\&\quad = { \mathchoice{\dfrac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }}{\dfrac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }}{\frac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }}{\frac{ \mathcal {A}\big (\alpha ,\beta ,\gamma \big ) }{ \sqrt{ \omega _1 \omega _2 } }} } \cdot \mathrm {e}^{ \pm 2{\mathrm i}\pi \big (\alpha ^{\star } w + \gamma ^{\star } u \big ) } D_{\beta ^{\star }}\big ( \omega _1 \omega _2 (u-w) \big ) \end{aligned}$$

(262)

provided that that $\alpha +\beta +\gamma =-{\mathrm i}\Omega$ and

$$\begin{aligned}&\int \limits _{ \mathbb {R}}^{} D_{\alpha }\big ( \omega _1 \omega _2 (x-u) \big ) \cdot D_{\beta }\big ( \omega _1 \omega _2 (x-v)\big ) \cdot D_{\gamma }\big ( \omega _1 \omega _2 (x-w) \big ) \cdot \mathrm {e}^{ \pm 2 {\mathrm i}\pi \delta x} \cdot \mathrm {d}x \nonumber \\&\quad = \mathcal {A}\big (\alpha ,\beta ,\gamma ,\delta \big ) \mathrm {e}^{ \pm 2 {\mathrm i}\pi \big (\alpha +\beta +{\mathrm i}{ \mathchoice{\dfrac{\Omega }{2}}{\dfrac{\Omega }{2}}{\frac{\Omega }{2}}{\frac{\Omega }{2}} }\big ) w } D_{\alpha + \beta + {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{ 2 }}{\dfrac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }} } } \big ( u - v \big ) \nonumber \\&\qquad \times \int \limits _{ \mathbb {R}}^{} D_{\alpha ^{\star }}\big ( \omega _1 \omega _2 (x-v) \big ) \cdot D_{\beta ^{\star }}\big ( \omega _1 \omega _2 (x-u) \big ) \nonumber \\&\qquad \times D_{\delta ^{\star }}\big ( \omega _1 \omega _2 (x-w) \big ) \cdot \mathrm {e}^{ \pm 2 {\mathrm i}\pi \gamma ^{\star } x} \cdot \mathrm {d}x \end{aligned}$$

(263)

provided that $\alpha +\beta +\gamma +\delta \, = \, -{\mathrm i}\Omega$.

The three-term integral relation can be recast in an operator form as

$$\begin{aligned} D_u( \varvec{ \texttt {p} } )\cdot D_{u+v}(\omega _1\omega _2 \varvec{ \texttt {x} } ) \cdot D_v( \varvec{ \texttt {p} } ) \; = \; D_v(\omega _1\omega _2 \varvec{ \texttt {x} } ) \cdot D_{u+v}( \varvec{ \texttt {p} } ) \cdot D_u(\omega _1\omega _2 \varvec{ \texttt {x} } ) \end{aligned}$$

(264)

see e.g. [17], whereas its degenerate form can be recast as

$$\begin{aligned} D_{\alpha }( \varvec{ \texttt {p} } ) \cdot \mathrm {e}^{\pm 2{\mathrm i}\pi \beta \varvec{ \texttt {x} } } \cdot D_{\gamma }( \varvec{ \texttt {p} } ) \; = \;\mathrm {e}^{\pm 2{\mathrm i}\pi \gamma \varvec{ \texttt {x} } } \cdot D_{\beta }( \varvec{ \texttt {p} } ) \cdot \mathrm {e}^{\pm 2{\mathrm i}\pi \alpha \varvec{ \texttt {x} } } \end{aligned}$$

(265)

where $\alpha ,\beta ,\gamma$ fulfil the constraint $\beta =\alpha +\gamma$.

Let $y_{\pm }$, $t_{\pm }$ be parameters as in (21). Then, the integral identity involving four D functions can be recast in the operator form as

$$\begin{aligned}&D_{ y_-}\big ( \varvec{ \texttt {p} } _2\big )\cdot D_{ y_+}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{23}\big )\cdot D_{ t_-}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) \cdot D_{ t_+}\big ( \varvec{ \texttt {p} } _2\big ) \nonumber \\&\quad = D_{y_-^{\star } + t_- + {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{ 2 }}{\dfrac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }} } } \big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) \cdot D_{ t_-}\big ( \varvec{ \texttt {p} } _2\big )\cdot D_{ t_+}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{23}\big )\cdot D_{ y_-}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) \cdot D_{ y_+}\big ( \varvec{ \texttt {p} } _2\big )\nonumber \\&\qquad \times D_{y_-^{\star } + t_- + {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{ 2 }}{\dfrac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }} } }^{ - 1 } \big (\omega _1\omega _2 \varvec{ \texttt {x} } _{32}\big ) \;. \end{aligned}$$

(266)

Likewise, its exponential degenerate form can be recast as

$$\begin{aligned}&\mathrm {e}^{2{\mathrm i}\pi y_- \varvec{ \texttt {x} } _1} \cdot D_{ y_-}\big ( \varvec{ \texttt {p} } _1\big )\cdot D_{ y_+}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) \cdot \mathrm {e}^{2{\mathrm i}\pi t_- \varvec{ \texttt {x} } _1} \cdot D_{ t_+}\big ( \varvec{ \texttt {p} } _1\big ) \nonumber \\&\quad = \mathrm {e}^{2{\mathrm i}\pi t_- \varvec{ \texttt {x} } _1} \cdot D_{ t_-}\big ( \varvec{ \texttt {p} } _1\big )\cdot D_{ t_+}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) \cdot \mathrm {e}^{2{\mathrm i}\pi y_- \varvec{ \texttt {x} } _1} \cdot D_{ y_+}\big ( \varvec{ \texttt {p} } _1\big )\nonumber \\&\qquad \times D_{y_-^{\star } + t_- + {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{ 2 }}{\dfrac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }} } }^{ - 1 } \big (\omega _1\omega _2 \varvec{ \texttt {x} } _{21}\big ) \;. \end{aligned}$$

(267)

Finally, the exponential degenerate form of the four D function integrals (263) can be also recast as

$$\begin{aligned}&\mathcal {A}(t_+) \, D_{ y_-}\big ( \varvec{ \texttt {p} } _2\big )\cdot D_{ y_+}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{23}\big )\cdot \mathrm {e}^{ \pm 2{\mathrm i}\pi (t_+^{\star } \varvec{ \texttt {x} } _2+y_+^{\star } \varvec{ \texttt {x} } _3)} \cdot D_{ t_-}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) \nonumber \\&\quad = \mathcal {A}(y_+) \, D_{y_-^{\star } + t_- + {\mathrm i}{ \mathchoice{\dfrac{ \Omega }{ 2 }}{\dfrac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }}{\frac{ \Omega }{ 2 }} } } \big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) D_{ t_-}\big ( \varvec{ \texttt {p} } _2\big )\cdot D_{ t_+}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{23}\big )\nonumber \\&\qquad \times \mathrm {e}^{ \pm 2{\mathrm i}\pi (y_+^{\star } \varvec{ \texttt {x} } _2+t_+^{\star } \varvec{ \texttt {x} } _3)} \cdot D_{ y_-}\big (\omega _1\omega _2 \varvec{ \texttt {x} } _{12}\big ) \;. \end{aligned}$$

(268)