Appendices
3.1.1 Appendix 1: The Proof of (3.2.28)
Here we estimate the conjugate \(\overline{C_{1}(j^{{\prime }},\lambda _{j,\beta })}\) of \(C_{1}(j^{{\prime }},\lambda _{j,\beta }),\) namely we prove that
$$\begin{aligned} \sum _{(j_{1},\beta _{1})\in Q(\rho ^{\alpha },9r)}\dfrac{\overline{ A(j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1}})\overline{ A(j^{{\prime }}+j_{1},\beta +\beta _{1},j,\beta })}{\lambda _{j,\beta }-\lambda _{j^{{\prime }}+j_{1},\beta +\beta _{1}}}=O(\rho ^{-2a}r^{2}), \end{aligned}$$
(3.5.1)
[see (3.2.26)], where
$$\begin{aligned} Q(\rho ^{\alpha },9r)=\{(j_{1},\beta _{1}):\vert j_{1}\delta \vert <9r,0<\vert \beta _{1}\vert <\rho ^{\alpha }\},j\in S_{1}(\rho ),\vert j^{{\prime }}\delta \vert <r,r=O(\rho ^{\frac{1}{2}\alpha _{2}}). \end{aligned}$$
The conditions on indices \(j^{{\prime }}\), \(j_{1},\)
\(j\) and (3.2.20) imply that
$$\begin{aligned} \mu _{j^{{\prime }}+j_{1}}=O(r^{2}),\mu _{j}=O(r^{2}). \end{aligned}$$
These with \(\beta \notin V_{\beta _{1}}^{\delta }(\rho ^{a})))\), where \( \beta _{1}\in \Gamma _{\delta }(p\rho ^{\alpha }),\) [see (3.2.9)] give
$$\begin{aligned} \lambda _{j,\beta }-\lambda _{j^{{\prime }}+j_{1},\beta +\beta _{1}}=-2\langle \beta ,\beta _{1}\rangle +O(r^{2}),\vert \langle \beta ,\beta _{1}\rangle \vert >\frac{1}{3}\rho ^{a}. \end{aligned}$$
(3.5.2)
Using this, (3.2.15) and (3.5.1), we get
$$\begin{aligned} \overline{C_{1}(j^{{\prime }},\lambda _{j,\beta })}=\sum _{\beta _{1}}\dfrac{ C^{{\prime }}}{-2\langle \beta ,\beta _{1}\rangle }+O(\rho ^{-2a}r^{2}), \end{aligned}$$
(3.5.3)
where
$$\begin{aligned} C^{{\prime }}=\sum _{j_{1}}\overline{A(j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1}})\overline{A(j^{{\prime }}+j_{1},\beta +\beta _{1},j,\beta }). \end{aligned}$$
In Chap. 2, we proved that [see (2.3.7), (2.3.21), Lemma 2.3.3]
$$\begin{aligned}&\overline{A(j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1}}) =\sum \limits _{n_{1}:(n_{1},\beta _{1})\in \Gamma ^{{\prime }}(\rho ^{\alpha })}c(n_{1},\beta _{1})a(n_{1},\beta _{1},j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1}), \nonumber \\&\quad \overline{A(j^{{\prime }}+j_{1},\beta +\beta _{1},j,\beta }) =\sum \limits _{n_{2}:(n_{2},-\beta _{1})\in \Gamma ^{{\prime }}(\rho ^{\alpha })}c(n_{2},-\beta _{1})a(n_{2},-\beta _{1},j^{{\prime }}+j_{1},\beta +\beta _{1},j,\beta ),\nonumber \\&\quad \Gamma ^{{\prime }}(\rho ^{\alpha })=\{(n_{1},\beta _{1}):\beta _{1}\in \Gamma _{\delta }\backslash 0,n_{1}\in \mathbb {Z},\beta _{1}+(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\delta \in \Gamma (\rho ^{\alpha })\}, \end{aligned}$$
(3.5.4)
$$\begin{aligned}&c(n_{1},\beta _{1})=q_{\gamma _{1}},\gamma _{1}=\beta _{1}+(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\delta \in \Gamma (\rho ^{\alpha }), \nonumber \\&\quad a(n_{1},\beta _{1},j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1})=(e^{i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j^{{\prime }},v(\beta )}(\zeta ),\varphi _{j^{{\prime }}+j_{1},v(\beta +\beta _{1})}(\zeta )), \end{aligned}$$
(3.5.5)
$$\begin{aligned} a(n_{2},-\beta _{1},j^{{\prime }}&+j_{1},\beta +\beta _{1},j,\beta )=(e^{i(n_{2}-(2\pi )^{-1}\langle -\beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j^{{\prime }}+j_{1},v(\beta +\beta _{1})}(\zeta ),\varphi _{j,v(\beta )}(\zeta ))\nonumber \\&\quad =(\varphi _{j^{{\prime }}+j_{1},v(\beta +\beta _{1})}(\zeta ),e^{-i(n_{2}-(2\pi )^{-1}\langle -\beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j,v(\beta )}(\zeta )) \nonumber \\&\quad =\overline{(e^{-i(n_{2}-(2\pi )^{-1}\langle -\beta _{1},\delta ^{*}\rangle \zeta }\varphi _{j,v(\beta )}(\zeta ),\varphi _{j^{{\prime }}+j_{1},v(\beta +\beta _{1})}(\zeta ))}, \end{aligned}$$
(3.5.6)
where \(\delta ^{*}\) is the element of \(\Omega \) satisfying \(\langle \delta ^{*},\delta \rangle =2\pi .\)
Now, to estimate the right-hand side of (3.5.3) we prove that
$$\begin{aligned}&\sum _{j_{1}}a(n_{1},\beta _{1},j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1})a(n_{2},-\beta _{1},j^{{\prime }}+j_{1},\beta +\beta _{1},j,\beta ) \\&=a(n_{1}+n_{2},0,j^{{\prime }},\beta ,j,\beta )+O(\rho ^{-p\alpha }). \nonumber \end{aligned}$$
(3.5.7)
By definition, we have
$$\begin{aligned}&a(n_{1}+n_{2},0,j^{{\prime }},\beta ,j,\beta )=(e^{i(n_{1}+n_{2})\zeta }\varphi _{j^{{\prime }},v(\beta )}(\zeta ),\varphi _{j,v(\beta )}(\zeta ))\nonumber \\&\quad = (e^{i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j^{{\prime }},v(\beta )}(\zeta ),e^{-i(n_{2}-(2\pi )^{-1}\langle -\beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j,v(\beta )}(\zeta )). \end{aligned}$$
This, (3.5.6), and the following formulas
$$\begin{aligned}&e^{i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j^{{\prime }},v(\beta )}(\zeta ) \nonumber \\&\qquad \qquad =\sum _{\vert j_{1}\delta \vert <9r}a(n_{1},\beta _{1},j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1})\varphi _{j^{{\prime }}+j_{1},v(\beta +\beta _{1})}(\zeta )+O(\rho ^{-p\alpha }), \nonumber \\&e^{-i(n_{2}-(2\pi )^{-1}\langle -\beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j,v(\beta )}(\zeta ) \nonumber \\&\qquad \qquad =\sum _{\vert j_{1}\delta \vert <9r}\overline{a(n_{2},-\beta _{1},j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1})}\varphi _{j^{{\prime }}+j_{1},v(\beta +\beta _{1})}+O(\rho ^{-p\alpha }),\nonumber \\&\quad \sum \limits _{j_{_{1}}}\vert a(n_{1},\beta _{1},j^{{\prime }},\beta ,j^{{\prime }}+j_{1},\beta +\beta _{1})\vert =O(1) \end{aligned}$$
(3.5.8)
[see (2.3.16), (2.3.17) of Chap. 2) give the proof of (3.5.7). Now from (3.5.7), (3.5.4) and (3.5.3) we obtain
$$\begin{aligned}&C^{{\prime }}=\sum \limits _{n_{1}}\sum \limits _{n_{2}}(c(n_{1},\beta _{1})c(n_{2},-\beta _{1})a(n_{1}+n_{2},0,j^{{\prime }},\beta ,j,\beta )+O(\rho ^{-p\alpha })),\nonumber \\&\quad \overline{C_{1}(j^{{\prime }},\lambda _{j,\beta })}=\sum _{\beta _{1}}\text { }\sum \limits _{n_{1}}\sum \limits _{n_{2}}C_{1}^{{\prime }}(\beta _{1},n_{1},n_{2})+O(\rho ^{-2a}r^{2}), \end{aligned}$$
where
$$\begin{aligned} C_{1}^{{\prime }}(\beta _{1},n_{1},n_{2})=\dfrac{c(n_{1},\beta _{1})c(n_{2},-\beta _{1})a(n_{1}+n_{2},0,j^{{\prime }},\beta ,j,\beta )}{ -2\langle \beta ,\beta _{1}\rangle }. \end{aligned}$$
It is clear that
$$\begin{aligned} C_{1}^{{\prime }}(\beta _{1},n_{1},n_{2})+C_{1}^{{\prime }}(-\beta _{1},n_{2},n_{1})=0. \end{aligned}$$
(3.5.9)
Therefore
$$\begin{aligned} \overline{C_{1}(j^{{\prime }},\lambda _{j,\beta })}=O(\rho ^{-2a}r^{2}). \end{aligned}$$
3.1.2 Appendix 2: The Proof of (3.2.35)
Arguing as in the proof of (3.2.27), we see that
$$\begin{aligned} C_{2}(\Lambda _{j,\beta })=C_{2}(\lambda _{j,\beta })+O(\rho ^{-3a}) \end{aligned}$$
and by (3.5.4)
$$\begin{aligned} \overline{C_{2}(\lambda _{j,\beta })}&=\sum _{\beta _{1},\beta _{2}}(\sum \limits _{n_{1},n_{2},n_{3}}(\sum _{j_{1},j_{2}}\dfrac{c(n_{1},\beta _{1})c(n_{2},\beta _{2})c(n_{3},-\beta _{1}-\beta _{2})}{(\lambda _{j,\beta }-\lambda _{j(1),\beta (1)})(\lambda _{j,\beta }-\lambda _{j(2),\beta (2)})}a(n_{1},\beta _{1},j,\beta ,j(1),\beta (1)) \\&\times a(n_{2},\beta _{2},j(1),\beta (1),j(2),\beta (2))a(n_{3},-\beta _{1}-\beta _{2},j(2),\beta (2),j,\beta ), \end{aligned}$$
where
$$\begin{aligned} (j_{1},\beta _{1})\in Q(\rho ^{\alpha },9r_{1}),(j_{2},\beta _{2})\in Q(\rho ^{\alpha },90r_{1}),j\in S_{1},\beta _{1}+\beta _{2}\ne 0. \end{aligned}$$
Applying (3.5.7) two times and using (3.5.8), we get
$$\begin{aligned}&\sum _{j_{1}}a(n_{1},\beta _{1},j,\beta ,j(1),\beta (1))(\sum _{j_{2}}a(n_{2}, \beta _{2},j(1),\beta (1),j(2),\beta (2))a(n_{3},-\beta _{1}- \beta _{2},j(2),\beta (2),j,\beta ))\nonumber \\&\quad =\sum _{j_{1}}a(n_{1},\beta _{1},j,\beta ,j(1),\beta (1))(a(n_{2}+n_{3},- \beta _{1},j(1),\beta (1),j,\beta )+O(\rho ^{-p\alpha }))\nonumber \\&\quad =a(n_{1}+n_{2}+n_{3},0,j,\beta ,j,\beta )+O(\rho ^{-p\alpha }). \end{aligned}$$
Using this in the above expression for \(C_{2}(\lambda _{j,\beta })\) and taking into account that
$$\begin{aligned} \lambda _{j,\beta }-\lambda _{j(1),\beta (1)}&=-2\langle \beta ,\beta _{1}\rangle +O(\rho ^{2\alpha _{1}}),\vert \langle \beta ,\beta _{1}\rangle \vert >\frac{1}{3}\rho ^{a}, \\ \lambda _{j,\beta }-\lambda _{j(2)\beta (2)}&=-2\langle \beta ,\beta _{1}+\beta _{2}\rangle +O(\rho ^{2\alpha _{1}}),\vert \langle \beta ,\beta _{1}+\beta _{2}\rangle \vert >\frac{1}{3}\rho ^{a}, \end{aligned}$$
which can be proved as (3.5.2), we have
$$\begin{aligned}&C_{2}(\lambda _{j,\beta })=O(\rho ^{-3a+2\alpha _{1}})\nonumber \\&\quad +\sum _{\beta _{1},\beta _{2}}\sum \limits _{n_{1},n_{2},n_{3}}\frac{ c(n_{1},\beta _{1})c(n_{2},\beta _{2})c(n_{3},-\beta _{1}-\beta _{2})a(n_{1}+n_{2}+n_{3},0,j,\beta ,j,\beta )}{4\langle \beta ,\beta _{1}\rangle \langle \beta ,\beta _{1}+\beta _{2}\rangle }. \end{aligned}$$
Grouping the terms with the equal multiplicands
$$\begin{aligned}&c(n_{1},\beta _{1})c(n_{2},\beta _{2})c(n_{3},-\beta _{1}-\beta _{2}), c(n_{2},\beta _{2})c(n_{1},\beta _{1})c(n_{3},-\beta _{1}-\beta _{2}), \\&c(n_{1},\beta _{1})c(n_{3},-\beta _{1}-\beta _{2})c(n_{2},\beta _{2}), c(n_{2},\beta _{2})c(n_{3},-\beta _{1}-\beta _{2})c(n_{1},\beta _{1}), \\&c(n_{3},-\beta _{1}-\beta _{2})c(n_{1},\beta _{1})c(n_{2},\beta _{2}), c(n_{3},-\beta _{1}-\beta _{2})c(n_{2},\beta _{2})c(n_{1},\beta _{1}) \end{aligned}$$
and using the obvious equality
$$\begin{aligned}&\frac{1}{\langle \beta ,\beta _{1}\rangle \langle \beta ,\beta _{1}+\beta _{2}\rangle }+\frac{1}{\langle \beta ,\beta _{2}\rangle \langle \beta ,\beta _{2}+\beta _{1}\rangle }+\frac{1}{\langle \beta ,\beta _{1}\rangle \langle \beta ,-\beta _{2}\rangle }\nonumber \\&\quad + \frac{1}{\langle \beta ,\beta _{2}\rangle \langle \beta -,\beta _{1}\rangle } +\frac{1}{\langle \beta ,-\beta _{1}-\beta _{2}\rangle \langle \beta ,-\beta _{2}\rangle }+\frac{1}{\langle \beta ,-\beta _{1}-\beta _{2}\rangle \langle \beta ,-\beta _{1}\rangle }=0, \end{aligned}$$
we see that
$$\begin{aligned} C_{2}(\lambda _{j,\beta })=O(\rho ^{-3a+2\alpha _{1}}). \end{aligned}$$
3.1.3 Appendix 3: The Proof of (3.2.34)
By (3.2.27) we have
$$\begin{aligned} C_{1}(\Lambda _{j,\beta })=C_{1}(\lambda _{j,\beta })+O(\rho ^{-3a}). \end{aligned}$$
Therefore, we need to prove that
$$\begin{aligned} \overline{C_{1}(\lambda _{j,\beta })}=\frac{1}{4}\int _{F}\left| f_{\delta ,\beta +\tau }(x)\right| ^{2}\left| \varphi _{j,v}^{\delta }(\langle \delta ,x\rangle )\right| ^{2}dx+O(\rho ^{-3a+2\alpha _{1}}), \end{aligned}$$
where
$$\begin{aligned}&\overline{C_{1}(\lambda _{j,\beta })}\equiv \sum _{\beta _{1}} \sum _{j_{1}}\dfrac{\overline{A(j,\beta ,j+j_{1},\beta +\beta _{1}})\overline{ A(j+j_{1},\beta +\beta _{1},j,\beta })}{\lambda _{j,\beta }-\lambda _{j+j_{1},\beta +\beta _{1}}},\nonumber \\&\quad (j_{1},\beta _{1})\in Q(\rho ^{\alpha },9r_{1}),j\in S_{1}, \end{aligned}$$
and by (3.5.4)
$$\begin{aligned}&\overline{C_{1}(\lambda _{j,\beta })}=\sum _{\beta _{1}} \sum \limits _{n_{1}:(n_{1},\beta _{1})\in \Gamma ^{{\prime }}(\rho ^{\alpha })}\sum \limits _{n_{2}:(n_{2},-\beta _{1})\in \Gamma ^{{\prime }}(\rho ^{\alpha })}\sum _{j_{1}}\frac{c(n_{1},\beta _{1})c(n_{2},-\beta _{1})}{\lambda _{j,\beta }-\lambda _{j+j_{1},\beta +\beta _{1}}}\nonumber \\&\qquad \qquad \qquad \times a(n_{1},\beta _{1},j,\beta ,j+j_{1},\beta +\beta _{1})a(n_{2},-\beta _{1},j+j_{1},\beta +\beta _{1},j,\beta ). \end{aligned}$$
Replacing \(\lambda _{j,\beta }-\lambda _{j+j_{1},\beta +\beta _{1}}\) by
$$\begin{aligned} -(2\langle \beta +\tau ,\beta _{1}\rangle +\vert \beta _{1}\vert ^{2}+\mu _{j+j_{1}}(v(\beta +\beta _{1}))-\mu _{j}(v(\beta ))) \end{aligned}$$
and using (3.5.7) for \(j^{{\prime }}=j,\) we have
$$\begin{aligned}&\overline{C_{1}(j,\lambda _{j,\beta })}=\sum _{\beta _{1}} \sum \limits _{n_{1}}\sum \limits _{n_{2}}\frac{c(n_{1},\beta _{1})c(n_{2},-\beta _{1})a(n_{1}+n_{2},0,j,\beta ,j,\beta )}{-2\langle \beta +\tau ,\beta _{1}\rangle }\nonumber \\&\quad + \sum _{\beta _{1}}\sum \limits _{n_{1}}\sum \limits _{n_{2}}\text { }\sum _{j_{1}}\frac{c(n_{1},\beta _{1})c(n_{2},-\beta _{1})a(n_{1},\beta _{1},j,\beta ,j+j_{1},\beta +\beta _{1})}{2\langle \beta +\tau ,\beta _{1}\rangle (2\langle \beta +\tau ,\beta _{1}\rangle +\vert \beta _{1}\vert ^{2}+\mu _{j+j_{1}}-\mu _{j})}\nonumber \\&\quad \times a(n_{2},-\beta _{1},j+j_{1},\beta +\beta _{1},j,\beta )(\vert \beta _{1}\vert ^{2}+\mu _{j+j_{1}}(v(\beta +\beta _{1}))-\mu _{j}(v(\beta ))). \end{aligned}$$
The formula (3.5.9) shows that the first summation of the right-hand side of this equality is zero. Thus we need to estimate the second sum. For this we use the following relation
$$\begin{aligned}&\mu _{j+j_{1}}(v(\beta +\beta _{1}))a(n_{1},\beta _{1},j,\beta ,j+j_{1},\beta +\beta _{1})=(e^{i(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })\zeta }\varphi _{j,v(\beta )}(\zeta ),T_{v}\varphi _{j+j_{1},v(\beta +\beta _{1})}(\zeta ))\nonumber \\&\quad =(T_{v}(e^{i(n_{1}-(2\pi )^{-1}(\beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j,v(\beta )}(\zeta )),\varphi _{j+j_{1},v(\beta +\beta _{1})}(\zeta )\nonumber \\&\quad =(\vert n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle \vert ^{2}\vert \delta \vert ^{2}+\mu _{j}(v))(e^{i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j,v(\beta )}(\zeta ),\varphi _{j+j_{1},v(\beta +\beta _{1})}(\zeta ))\nonumber \\&\quad \quad \quad -2i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\vert \delta \vert ^{2}(e^{i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j,v(\beta )}^{{\prime }}(\zeta ),\varphi _{j+j_{1},v(\beta +\beta _{1})}(\zeta )). \end{aligned}$$
Using this, (3.5.7), and the formula
$$\begin{aligned}&\sum _{j_{1}}(e^{i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\zeta }\varphi _{j,v(\beta )}^{{\prime }}(\zeta )),\varphi _{j+j_{1},v(\beta +\beta _{1})}(\zeta ))a(n_{2},-\beta _{1},j+j_{1},\beta +\beta _{1},j,\beta ) \\&=(e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}^{{\prime }}(\zeta )),\varphi _{j,v(\beta )}(\zeta ))+O(\rho ^{-p\alpha }) \end{aligned}$$
which can be proved as (3.5.7), we obtain
$$\begin{aligned}&\sum _{j_{1}}\mu _{j+j_{1}}(v(\beta +\beta _{1})a(n_{1},\beta _{1},j,\beta ,j+j_{1},\beta +\beta _{1})a(n_{2},-\beta _{1},j+j_{1},\beta +\beta _{1},j,\beta )\nonumber \\&\quad \quad =(\vert n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle \vert ^{2})\vert \delta \vert ^{2}+\mu _{j}(v)a(n_{1}+n_{2},0,j,\beta ,j,\beta )\nonumber \\&\quad \qquad - 2i(n_{1}-(2\pi )^{-1}\langle \beta _{1},\delta ^{*}\rangle )\vert \delta \vert ^{2}(e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}^{{\prime }}(\zeta ),\varphi _{j,v(\beta )}(\zeta )). \end{aligned}$$
(3.5.10)
Here the last multiplicand can be estimated as follows
$$\begin{aligned}&\ \mu _{j}(v)(\varphi _{j,v(\beta )}(\zeta ),e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}(\zeta ))=(\varphi _{j,v(\beta )}(\zeta ),T_{v}(e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}(\zeta )))\nonumber \\&\quad =(n_{1}+n_{2})^{2}\vert \delta \vert ^{2}(\varphi _{j,v(\beta )}(\zeta ),e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}(\zeta ))\nonumber \\&\quad \quad +2i(n_{1}+n_{2})\vert \delta \vert ^{2}(\varphi _{j,v(\beta )}(\zeta ),e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}^{{\prime }}(\zeta ))+\mu _{j}(v)(\varphi _{j,v(\beta )},e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}),\nonumber \\&\quad (e^{i(n_{1}+n_{2})\zeta }\varphi _{j,v(\beta )}^{{\prime }}(\zeta )),\varphi _{j,v(\beta )}(\zeta ))=\frac{n_{1}+n_{2}}{2i}(e^{i(n_{1}+n_{2}) \zeta }\varphi _{j,v(\beta )}(\zeta )),\varphi _{j,v(\beta )}(\zeta )). \end{aligned}$$
Using this, (3.5.10), and (3.5.7), we get
$$\begin{aligned}&\sum _{j_{1}}(a(n_{1},\beta _{1},j,\beta ,j+j_{1},\beta +\beta _{1})a(n_{2},-\beta _{1},j+j_{1},\beta +\beta _{1},j,\beta ))\nonumber \\&\quad \times (\vert \beta _{1}\vert ^{2}+\mu _{j+j_{1}}(v(\beta +\beta _{1}))-\mu _{j}(v(\beta )))=a(n_{1}+n_{2},0,j,\beta ,j,\beta )\nonumber \\&\quad \times (\vert \beta _{1}\vert ^{2}+\vert n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi }\vert ^{2}\vert \delta \vert ^{2}-(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })\vert \delta \vert ^{2}(n_{1}+n_{2}))\nonumber \\&=(\vert \beta _{1}\vert ^{2}+\vert \delta \vert ^{2}(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })(-n_{2}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi }))a(n_{1}+n_{2},0,j,\beta ,j,\beta ). \end{aligned}$$
Thus
$$\begin{aligned} \overline{C_{1}(j,\lambda _{j,\beta })}=C+O(\rho ^{-3a+2\alpha _{1}}), \end{aligned}$$
where
$$\begin{aligned}&C=\sum _{\beta _{1},n_{1},n_{2}}\frac{c(n_{1},\beta _{1})c(n_{2},-\beta _{1})a(n_{1}+n_{2},0,j,\beta ,j,\beta )}{4\vert \langle \beta +\tau ,\beta _{1}\rangle \vert ^{2}}\nonumber \\&\quad \quad \times (\vert \beta _{1}\vert ^{2}+(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })(-n_{2}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{ 2\pi })\vert \delta \vert ^{2}). \end{aligned}$$
(3.5.11)
Now we consider
$$\begin{aligned} \int _{F}\left| f_{\delta ,\beta +\tau }(x)\right| ^{2}\left| \varphi _{n,v}(\langle \delta ,x\rangle )\right| ^{2}dx, \end{aligned}$$
where \(f_{\delta ,\beta +\tau }\) is defined in (3.1.12). By (3.5.5)
$$\begin{aligned} f_{\delta ,\beta +\tau }(x)=\sum _{(n_{1},\beta _{1})\in \Gamma _{\delta }^{{\prime }}(\rho ^{\alpha })}\frac{\beta _{1}+(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })\delta }{\langle \beta +\tau ,\beta _{1}\rangle }c(n_{1},\beta _{1})e^{i\langle \beta _{1}+(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })\delta ,x\rangle }. \end{aligned}$$
Here \(f_{\delta ,\beta +\tau }(x)\) is a vector of \(\mathbb {R}^{d}\). Using \( \langle \beta ,\delta \rangle =0\) for \(\beta \in \Gamma _{\delta },\) we obtain
$$\begin{aligned}&\left| f_{\delta ,\beta +\tau }(x)\right| ^{2}=\sum _{(n_{1},\beta _{1}),(n_{2},\beta _{2})\in \Gamma _{\delta }^{{\prime }}(\rho ^{\alpha })} \frac{\langle \beta _{1},\beta _{2}\rangle +(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })(n_{2}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })\vert \delta \vert ^{2}}{\langle \beta +\tau ,\beta _{1}\rangle \langle \beta +\tau ,\beta _{2}\rangle }\nonumber \\&\qquad \qquad \qquad \qquad \times c(n_{1},\beta _{1})c(-n_{2},-\beta _{2})e^{i\langle \beta _{1}-\beta _{2}+(n_{1}-n_{2}-(2\pi )^{-1}\langle \beta _{1}-\beta _{2},\delta ^{*}\rangle )\delta ,x\rangle }. \end{aligned}$$
Since \(\varphi _{j,v}(\langle \delta ,x\rangle )\) is a function of \(\langle \delta ,x\rangle ,\) we have
$$\begin{aligned} \int _{F}e^{i\langle \beta _{1}-\beta _{2}+(n_{1}-n_{2}-(2\pi )^{-1}\langle \beta _{1}-\beta _{2},\delta ^{*}\rangle )\delta ,x\rangle }\left| \varphi _{j,v}(\langle \delta ,x\rangle )\right| ^{2}dx=0 \end{aligned}$$
for \(\beta _{1}\ne \beta _{2}\). Therefore
$$\begin{aligned}&\int _{F}\left| f_{\delta ,\beta +\tau }(x)\right| ^{2}\left| \varphi _{j,v}(\langle \delta ,x\rangle )\right| ^{2}dx=\sum _{\beta _{1},n_{1},n_{2}}\frac{c(n_{1},\beta _{1})c(-n_{2},-\beta _{1})}{\vert \langle \beta +\tau ,\beta _{1}\rangle \vert ^{2}}\nonumber \\&\qquad \qquad \qquad \times (\vert \beta _{1}\vert ^{2}+(n_{1}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{2\pi })(n_{2}-\frac{\langle \beta _{1},\delta ^{*}\rangle }{ 2\pi })\vert \delta \vert ^{2}a(n_{1}-n_{2},0,j,\beta ,j,\beta \rangle . \end{aligned}$$
Replacing \(n_{2}\) by \(-n_{2},\) we get
$$\begin{aligned} \int _{F}\left| f_{\delta ,\beta +\tau }(x)\right| ^{2}\left| \varphi _{n,v}(\langle \delta ,x\rangle )\right| ^{2}dx=4C \end{aligned}$$
[see (3.5.11)] and (3.2.34).
3.1.4 Appendix 4: Asymptotic Formulas
for \(T_{v}(Q)\)
It is well-known that the large eigenvalues of \(T_{0}(Q)\) lie in \(O(\frac{1}{ m^{4}})\) neighborhood of
$$\begin{aligned} \vert m\delta \vert +\frac{1}{16\pi \vert m\delta \vert ^{3}}\int _{0}^{2\pi }\left| Q(t)\right| ^{2}dt \end{aligned}$$
for the large values of \(m\) (see [Eas], p. 58). This formula yields the invariant (3.1.16). Using the asymptotic formulas
for solutions of the Sturm-Liouville equation (see [Eas], p. 63), one can easily obtain that
$$\begin{aligned} \varphi _{n,v}(\zeta )=e^{i(n+v)\zeta }(1+\frac{Q_{1}(\zeta )}{2i(n+v)\vert \delta \vert ^{2}}+\frac{Q(\zeta )-Q(0)-\frac{1}{2}Q_{1}^{2}(\zeta )}{ 4(n+v)^{2}\vert \delta \vert ^{4}})+O(\frac{1}{n^{3}})), \end{aligned}$$
where
$$\begin{aligned} Q_{1}(\zeta )=\int _{0}^{\zeta }Q(t)dt. \end{aligned}$$
From this, by direct calculations, we find \(A_{0}(\zeta ),\)
\(A_{1}(\zeta ),\)
\(A_{2}(\zeta )\) [see (3.1.6)] and then using these in (3.1.7), we get the invariant (3.1.15).
Now we consider the eigenfunction \(\varphi _{n,v}(\zeta )\) of \(T_{v}(p)\) in the case \(v\ne 0,\ \frac{1}{2}\) and
$$\begin{aligned} p(\zeta )=p_{1}e^{i\zeta }+p_{-1}e^{-i\zeta }. \end{aligned}$$
The eigenvalues and the eigenfunctions of \(T_{v}(0)\) are \(\ (n+v)^{2}\vert \delta \vert ^{2}\) and \(\ e^{i(n+v)\zeta }\), for \(n\in \mathbb {Z}\). Since the eigenvalues of \(T_{v}(p)\) are simple for \(v\ne 0,\ \frac{1}{2},\) by the well-known perturbation formula
$$\begin{aligned}&(\varphi _{n,v}(\zeta ),\ e^{i(n+v)\zeta })\varphi _{n,v}(\zeta )=e^{i(n+v)\zeta }\nonumber \\&\quad +\sum _{k=1,2,\ldots }\frac{(-1)^{k+1}}{2i\pi }\int \limits _{C}(T_{v}(0)-\lambda )^{-1}p(x)^{k}(T_{v}(0)-\lambda )^{-1}e^{i(n+v)\zeta }d\lambda , \end{aligned}$$
(3.5.12)
where \(C\) is a contour containing only the eigenvalue \((n+t)^{2}\vert \delta \vert ^{2}\). Using
$$\begin{aligned} (T_{v}(0)-\lambda )^{-1}e^{i(n+v)\zeta }=\frac{e^{i(n+v)\zeta }}{ (n+v)^{2}\vert \delta \vert ^{2}-\lambda }, \end{aligned}$$
we see that the \(k\)th (\(k=1,2,3,4\)) term \(F_{k}\) of the series (3.5.12) has the form
$$\begin{aligned}&F_{1}=\frac{1}{2i\pi }\int \limits _{C}\sum _{m=1,-1}\frac{p_{m}e^{i(n+m+v) \zeta }}{((n+v)^{2}\vert \delta \vert ^{2}-\lambda )((n+m+v)^{2}\vert \delta \vert ^{2}-\lambda )}d\lambda ,\nonumber \\&F_{2}=\frac{-1}{2i\pi }\int \limits _{C}\sum _{m,l=1,-1}\frac{ p_{m}p_{l}e^{i(n+m+l+v)\zeta }}{((n+v)^{2}\vert \delta \vert ^{2}-\lambda )}\nonumber \\&\qquad \qquad \times \frac{1}{((n+m+v)^{2}\vert \delta \vert ^{2}-\lambda )((n+m+l+v)^{2}\vert \delta \vert ^{2}-\lambda )}d\lambda ,\nonumber \\&F_{3}=\frac{1}{2i\pi }\int \limits _{C}\sum _{m,l,k=1,-1}\frac{ p_{m}p_{l}p_{k}e^{i(n+m+l+k+v)\zeta }}{((n+v)^{2}\vert \delta \vert ^{2}-\lambda )((n+m+v)^{2}\vert \delta \vert ^{2}-\lambda )}\nonumber \\&\qquad \qquad \times \frac{1}{((n+m+l+v)^{2}\vert \delta \vert ^{2}-\lambda )((n+m+l+k+v)^{2}\vert \delta \vert ^{2}-\lambda )}d\lambda ,\nonumber \\&F_{4}=\frac{-1}{2i\pi }\int \limits _{C}\sum _{m,l,k,r=1,-1}\frac{ p_{m}p_{l}p_{k}p_{r}e^{i(n+m+l+k+r+v)\zeta }}{((n+m+l+k+r+v)^{2}\vert \delta \vert ^{2}-\lambda )}\nonumber \\&\qquad \qquad \times \frac{1}{((n+m+v)^{2}\vert \delta \vert ^{2}-\lambda )((n+m+l+v)^{2}\vert \delta \vert ^{2}-\lambda )}\nonumber \\&\qquad \qquad \times \frac{1}{((n+m+l+k+v)^{2}\vert \delta \vert ^{2}-\lambda )((n+v)^{2}\vert \delta \vert ^{2}-\lambda )}d\lambda . \end{aligned}$$
Since the distance between \((n+v)^{2}\vert \delta \vert ^{2}\) and \( (n^{{\prime }}+v)^{2}\vert \delta \vert ^{2}\) for \(n^{{\prime }}\ne n\) is greater than \(c_{17}n\), we can choose the contour \(C\) such that
$$\begin{aligned} \frac{1}{\vert (n^{{\prime }}+v)^{2}\vert \delta \vert ^{2}-\lambda \vert }< \frac{c_{18}}{n},\,\forall \lambda \in C,\,\forall n^{{\prime }}\ne n \end{aligned}$$
and the length of \(C\) is less than \(c_{19}\). Therefore
$$\begin{aligned} F_{5}+F_{6}+\cdots =O(n^{-5}). \end{aligned}$$
Now we calculate the integrals in \(F_{1}\), \(F_{2}\), \(F_{3}\), \(F_{4}\) by the Cauchy integral formula and then decompose the obtained expression in power of \(\frac{1}{n}\). Then
$$\begin{aligned}&F_{1}=e^{i(n+v)\zeta }((p_{1}e^{i\zeta }-p_{-1}e^{-i\zeta })\frac{1}{\vert \delta \vert ^{2}}(\frac{-1}{2n}+\frac{v}{2n^{2}}-\frac{4v^{2}+1}{8n^{3}}+O( \frac{1}{n^{4}}))\nonumber \\&\qquad \qquad +\,(p_{1}e^{i\zeta }+p_{-1}e^{-i\zeta })\frac{1}{\vert \delta \vert ^{2}}(\frac{v }{4n^{2}}-\frac{v}{2n^{3}}+\frac{12v^{2}+1}{16n^{4}}+O(\frac{1}{n^{5}}))). \end{aligned}$$
Let \(F_{2,1}\) and \(F_{2,2}\) be the sum of the terms in \(F_{2}\) for which \( m+l=\pm 2\) and \(m+l=0\) respectively, i.e.,
$$\begin{aligned} F_{2}=F_{2,1}+F_{2,2}, \end{aligned}$$
where
$$\begin{aligned}&F_{2,1}=e^{i(n+v)\zeta }(((p_{1})^{2}e^{2i\zeta }+(p_{-1})^{2}e^{-2i\zeta }) \frac{1}{\vert \delta \vert ^{4}}(\frac{-1}{8n^{2}}+\frac{-v}{4n^{3}}-\frac{ 12v^{2}+7}{32n^{4}}+O(\frac{1}{n^{5}}))\nonumber \\&\qquad \quad \quad +\,((p_{1})^{2}e^{2i\zeta }-(p_{-1})^{2}e^{-2i\zeta })\frac{1}{\vert \delta \vert ^{4}}(\frac{-3}{16n^{3}}+O(\frac{1}{n^{4}}))),\nonumber \\&F_{2,2}=e^{i(n+v)\zeta }\left| p_{1}\right| ^{2}(\frac{c_{20}}{n^{2}} +\frac{c_{21}}{n^{3}}+\frac{c_{22}}{n^{4}}+O(\frac{1}{n^{5}})) \end{aligned}$$
and \(c_{20},c_{21},c_{22}\) are the known constants. Similarly,
$$\begin{aligned} F_{3}=F_{3,1}+F_{3,2}, \end{aligned}$$
where \(F_{3,1}\) and \(F_{3,2}\) are the sum of the terms in \(F_{3}\) for which \(m+l+k=\pm 3\) and \(m+l+k=\pm 1\) respectively. Hence
$$\begin{aligned}&F_{3,1}=e^{i(n+v)\zeta }((p_{1}^{3}e^{3i\zeta }-p_{-1}^{3}e^{-i\zeta })\frac{ 1}{\vert \delta \vert ^{6}}(\frac{-1}{48n^{3}}+O(\frac{1}{n^{4}}))\nonumber \\&\qquad \qquad +\,(p_{1}^{3}e^{3i\zeta }+p_{-1}^{3}e^{-3i\zeta })\frac{1}{\vert \delta \vert ^{6} }(\frac{1}{16n^{4}}+O(\frac{1}{n^{5}}))),\nonumber \\&F_{3,2}=e^{i(n+v)\zeta }((p_{1}e^{i\zeta }-p_{-1}e^{-i\zeta })\left| p_{1}\right| ^{2}(\frac{c_{23}}{n^{3}}+\frac{c_{24}}{n^{4}}+O(\frac{1}{ n^{5}}))\nonumber \\&\qquad \qquad +\,(p_{1}e^{i\zeta }+p_{-1}e^{-i\zeta })\left| p_{1}\right| ^{2}(\frac{ c_{25}}{n^{4}}+O(\frac{1}{n^{5}}))). \end{aligned}$$
Similarly
$$\begin{aligned} F_{4}=F_{4,1}+F_{4,2}+F_{4,3}, \end{aligned}$$
where \(F_{4,1}\), \(F_{4,2}\), \(F_{4,3}\) are the sum of the terms in \(F_{4}\) for which \(m+l+k+r=\pm 4\), \(m+l+k+r=\pm 2\), \(m+l+k+r=0\) respectively. Thus
$$\begin{aligned} F_{4,1}&=e^{i(n+v)\zeta }(p_{1}^{4}e^{4i\zeta }+p_{-1}^{4}e^{-4i\zeta })\frac{ 1}{\vert \delta \vert ^{8}}(\frac{1}{384n^{4}}+O(\frac{1}{n^{5}})),\nonumber \\ F_{4,2}&=e^{i(n+v)\zeta }(p_{1}^{2}e^{2i\zeta }+p_{-1}^{2}e^{-2i\zeta })\left| p_{1}\right| ^{2}(\frac{c_{26}}{n^{4}}+O(\frac{1}{n^{5}}))),\nonumber \\ F_{4,3}&=e^{i(n+v)\zeta }\left| p_{1}\right| ^{4}(\frac{c_{27}}{n^{4}} +O(\frac{1}{n^{5}}))). \end{aligned}$$
Since \(p_{-1}^{k}e^{-ik\zeta }\) is conjugate of \(p_{1}^{k}e^{ik\zeta },\) the real and imaginary parts of \(F_{k}e^{-i(n+v)\zeta }\) consist of terms with multiplicands
$$ \begin{aligned} p_{1}^{k}e^{ik\zeta }+p_{-1}^{k}e^{-ki\zeta }\quad \& \quad p_{1}^{k}e^{ik\zeta }-p_{-1}^{k}e^{-ik\zeta } \end{aligned}$$
respectively. Taking into account this and using the above estimations, we get
$$\begin{aligned} |(\varphi _{n,v},e^{i(n+v)\zeta })\varphi _{n,v}|^{2}&= 2(\sum _{k=1,2,3,4} \mathbf {Re(}F_{k})+\mathbf {Re(}F_{1}F_{2})+\mathbf {Re(} F_{1}F_{3}))+|F_{1}|^{2}+|F_{2}|^{2}+O(n^{-5})\nonumber \\&=1+\frac{1}{2n^{2}}\frac{1}{\vert \delta \vert ^{2}}(p_{1}e^{i\zeta }+p_{-1}e^{-i\zeta }+c_{28}|p_{1}|^{2})+\frac{1}{n^{3}}((p_{1}e^{i\zeta }+p_{-1}e^{-i\zeta })c_{29}\nonumber \\&\quad +c_{30}|p_{1}|^{2})+\frac{1}{n^{4}}((p_{1}e^{i\zeta }+p_{-1}e^{-i\zeta })c_{31}+c_{32}|p_{1}|^{2}+c_{33}|p_{1}|^{4}\nonumber \\&\quad +c_{34}|p_{1}|^{2}(p_{1}e^{i\zeta }+p_{-1}e^{-i\zeta })+(c_{35}+c_{36}|p_{1}|^{2})(p_{1}^{2}e^{2i\zeta }+p_{-1}^{2}e^{-2i\zeta }))\nonumber \\&\quad +O(\frac{1}{n^{5}}), \end{aligned}$$
where \(\mathbf {Re(}F)\) denotes the real part of \(F.\) On the other hand
$$\begin{aligned} |(\varphi _{n,v}(\zeta ),e^{i(n+v)\zeta })|^{2}=(c_{37}\frac{1}{n^{2}}+c_{38} \frac{1}{n^{3}}+c_{39}\frac{1}{n^{4}})|p_{1}|^{2}+c_{40}\frac{1}{n^{4}} |p_{1}|^{4}+O(\frac{1}{n^{5}}). \end{aligned}$$
These equalities imply (3.1.18). The invariant (3.1.19) is a consequence of (3.1.18), (3.1.16) and (3.1.7) for \(k=2,4\)
\(\blacksquare \)