Appendix
Proof of Theorem 3.1
For readers who are quite familiar with the analytic KAM scheme, this proof can be skipped since the structure is similar to that in [13]. But the estimates here are sharp compared with those in [13] (see Remark 3.3), so we prefer to provide the detailed proof for self-containedness.
Recall that \(sl(2,{{\mathbb {R}}})\) is isomorphic to su(1, 1), which consists of matrices of the form
$$\begin{aligned} \begin{pmatrix} it &{}\quad v\\ \bar{v} &{}\quad -it \end{pmatrix} \end{aligned}$$
with \(t\in {{\mathbb {R}}}\), \(v\in {{\mathbb {C}}}\). The isomorphism between them is given by \(A\rightarrow MAM^{-1}\), where
$$\begin{aligned} M=\frac{1}{1+i}\begin{pmatrix} 1 &{}\quad -i\\ 1 &{}\quad i \end{pmatrix} \end{aligned}$$
and a simple calculation yields
$$\begin{aligned} M\begin{pmatrix} x &{}\quad y+z\\ y-z &{}\quad -x \end{pmatrix}M^{-1}=\begin{pmatrix} iz &{}\quad x-iy\\ x+iy &{}\quad -iz \end{pmatrix}, \end{aligned}$$
where \(x,y,z\in {{\mathbb {R}}}\). SU(1, 1) is the corresponding Lie group of su(1, 1). We will prove this theorem in SU(1, 1), which is isomorphic to \(SL(2,{{\mathbb {R}}})\). \(\square \)
We distinguish two cases:
Non-resonant case For \(0<|n |\leqslant N=\frac{2}{r-r'} |\ln \epsilon |\), we have
$$\begin{aligned} |2\rho - \langle n,\alpha \rangle |\geqslant \epsilon ^{\sigma }; \end{aligned}$$
(5.1)
by (3.2) with \(D>\frac{2}{\sigma }\), we have
$$\begin{aligned} \left|\langle n,\alpha \rangle \right|\geqslant \frac{\kappa }{\left|n \right|^{\tau }}\geqslant \frac{\kappa }{\left|N \right|^{\tau }}\geqslant \epsilon ^{\frac{\sigma }{2}}\geqslant \epsilon ^{\sigma }. \end{aligned}$$
(5.2)
It is well known that (5.1) and (5.2) are the conditions which are used to overcome the small denominator problem in KAM theory.
Define
$$\begin{aligned} \Lambda _N= \left\{ f\in C^{\omega }_{r}({{\mathbb {T}}}^{d},su(1,1))\mid f(\theta )=\sum _{k\in {{\mathbb {Z}}}^{d},0<|k |<N}\hat{f}(k)e^{i\langle k,\theta \rangle }\right\} . \end{aligned}$$
(5.3)
Our goal is to solve the cohomological equation
$$\begin{aligned} Y(\theta +\alpha )A-AY(\theta )=A(-\mathcal {T}_Nf(\theta )+\hat{f}(0)), \end{aligned}$$
i.e.,
$$\begin{aligned} A^{-1}Y(\theta +\alpha )A-Y(\theta )=-\mathcal {T}_Nf(\theta )+\hat{f}(0). \end{aligned}$$
(5.4)
Here \(\mathcal {T}_N\) is the truncation operator such that
$$\begin{aligned} (\mathcal {T}_Nf)(\theta )=\sum _{k\in {{\mathbb {Z}}}^{d},|k |<N}\hat{f}(k)e^{i\langle k,\theta \rangle }. \end{aligned}$$
Take the Fourier transform for (5.4) and compare the corresponding Fourier coefficients of the two sides. By (5.1) (apply it twice to solve the off-diagonal) along with (5.2) (apply it once to solve the diagonal), we obtain that if \(Y\in \Lambda _N\), then
$$\begin{aligned} |Y(\theta )|_r \leqslant \epsilon ^{-3\sigma }|\mathcal {T}_Nf(\theta )-\hat{f}(0)|_r, \end{aligned}$$
which gives
$$\begin{aligned} |A^{-1}Y(\theta +\alpha )A-Y(\theta )|_r\geqslant \epsilon ^{3\sigma }|Y(\theta )|_r. \end{aligned}$$
(5.5)
Moreover, we have \(A^{-1}Y(\theta +\alpha )A \in \Lambda _N\) by (5.3). For \(\eta =\epsilon ^{3\sigma }\), we define \(\mathcal {B}_r^{nre}(\epsilon ^{3\sigma })\) by (3.1), then we have \(\Lambda _N \subset \mathcal {B}_r^{nre}(\epsilon ^{3\sigma })\).
Since \(\epsilon ^{3\sigma }\geqslant 13\Vert A\Vert ^2\epsilon ^{\frac{1}{2}}\) (it holds by \(\sigma \) being smaller than \(\frac{1}{6}\) and \(\tilde{D}\) depending on \(\sigma \)), by Lemma 3.1 we have \(Y\in \mathcal {B}_r\) and \(f^{re}\in \mathcal {B}_r^{re}(\epsilon ^{3\sigma })\) such that
$$\begin{aligned} e^{Y(\theta +\alpha )}(Ae^{f(\theta )})e^{-Y(\theta )}=Ae^{f^{re}(\theta )}, \end{aligned}$$
with \(|Y |_r\leqslant \epsilon ^{\frac{1}{2}}\) and
$$\begin{aligned} |f^{re}|_r\leqslant 2\epsilon . \end{aligned}$$
(5.6)
By (5.3)
$$\begin{aligned} (\mathcal {T}_N{f^{re}})(\theta )=\hat{f}^{re}(0), \ \ \Vert \hat{f}^{re}(0)\Vert \leqslant 2\epsilon , \end{aligned}$$
and
$$\begin{aligned} |(\mathcal {R}_N{f^{re}})(\theta )|_{r'}&= |\sum _{|n |>N}\hat{f}^{re}(n)e^{i\langle n,\theta \rangle }|_{r'}\nonumber \\&\leqslant 2\epsilon e^{-N(r-r')}(N)^{d}\nonumber \\&\leqslant 2\epsilon \cdot \epsilon ^{2}\cdot \frac{1}{4}\epsilon ^{-\sigma }\nonumber \\&=\frac{1}{2}\epsilon ^{3-\sigma }. \end{aligned}$$
(5.7)
Moreover, we can compute that
$$\begin{aligned} e^{\hat{f}^{re}(0)+\mathcal {R}_N{f^{re}}(\theta )}=e^{\hat{f}^{re}(0)}(Id+e^{-\hat{f}^{re}(0)}\mathcal {O}(\mathcal {R}_N{f^{re}}))=e^{\hat{f}^{re}(0)}e^{f_+(\theta )}, \end{aligned}$$
by (5.7), we have
$$\begin{aligned} |f_+(\theta )|_{r'}\leqslant 2|\mathcal {R}_N{f^{re}(\theta )}|_{r'} \leqslant \epsilon ^{3-\sigma }. \end{aligned}$$
Finally, if we denote
$$\begin{aligned} A_+=Ae^{\hat{f}^{re}(0)}, \end{aligned}$$
then we have
$$\begin{aligned} \Vert A_+-A\Vert \leqslant \Vert A\Vert \Vert Id-e^{\hat{f}^{re}(0)} \Vert \leqslant 2\Vert A\Vert \epsilon . \end{aligned}$$
Resonant case In fact, we only need to consider the case in which A is elliptic with eigenvalues \(\{e^{i\rho },e^{-i\rho }\}\) for \(\rho \in {{\mathbb {R}}}\backslash \{0\}\) since if \(\rho \in i{{\mathbb {R}}}\), then the non-resonant condition is always satisfied due to the Diophantine condition on \(\alpha \) and then it actually belongs to the non-resonant case.
Claim 2
\(n_*\) is the unique resonant site with
$$\begin{aligned} 0<|n_*|\leqslant N=\frac{2}{r-r'} |\ln \epsilon |. \end{aligned}$$
Proof
Indeed, if there exists \(n_{*}^{'}\ne n_*\) satisfying \(|2\rho - \langle n_{*}^{'},\alpha \rangle |< \epsilon ^{\sigma }\), then by the Diophantine condition of \(\alpha \), we have
$$\begin{aligned} \frac{\kappa }{|n_{*}^{'}-n_*|^{\tau }}\leqslant |\langle n_{*}^{'}-n_*,\alpha \rangle |< 2\epsilon ^{\sigma }, \end{aligned}$$
which implies that \(|n_{*}^{'} |>2^{-\frac{1}{\tau }}\kappa ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}-N> 2N^2.\) \(\square \)
Since we have
$$\begin{aligned} |2\rho - \langle n_*,\alpha \rangle |< \epsilon ^{\sigma }, \end{aligned}$$
(5.8)
the smallness condition on \(\epsilon \) implies that
$$\begin{aligned} |\ln \epsilon |^\tau \epsilon ^\sigma \leqslant \frac{\kappa (r-r')^\tau }{2^{\tau +1}}. \end{aligned}$$
Thus,
$$\begin{aligned} \frac{\kappa }{|n_*|^\tau } \leqslant |\langle n_*,\alpha \rangle |\leqslant \epsilon ^\sigma +2|\rho |\leqslant \frac{\kappa }{2|n_*|^\tau }+2|\rho |, \end{aligned}$$
which implies that
$$\begin{aligned} |\rho |\geqslant \frac{\kappa }{4|n_*|^{\tau }}. \end{aligned}$$
Then by Lemma 8.1 of Hou–You [22], one can find \(P\in SU(1,1)\) with
$$\begin{aligned} \Vert P \Vert \leqslant 2 \left( \frac{\Vert A \Vert }{|\rho |}\right) ^{\frac{1}{2}}\leqslant 4 \left( \frac{\Vert A\Vert }{\kappa }\right) ^{\frac{1}{2}}|n_*|^{\frac{\tau }{2}}, \end{aligned}$$
such that
$$\begin{aligned} PAP^{-1}=\begin{pmatrix} e^{i\rho } &{}\quad 0\\ 0 &{}\quad e^{-i\rho } \end{pmatrix}=A'. \end{aligned}$$
Denote \(g=PfP^{-1}\), by (3.2) we have:
$$\begin{aligned} \Vert P \Vert\leqslant & {} 4\left( \frac{\Vert A\Vert }{\kappa }\right) ^{\frac{1}{2}}|N |^{\frac{\tau }{2}}\leqslant 4 \left( \frac{\Vert A\Vert }{\kappa }\right) ^{\frac{1}{2}}\left( \frac{2}{r-r'} |\ln \epsilon |\right) ^{\frac{\tau }{2}}, \end{aligned}$$
(5.9)
$$\begin{aligned} |g |_r\leqslant & {} \Vert P \Vert ^2|f|_r \leqslant \frac{2^{4+\tau }\Vert A\Vert |\ln \epsilon |^{\tau }}{\kappa (r-r')^{\tau }}\times \epsilon :=\epsilon '. \end{aligned}$$
(5.10)
Now we define
$$\begin{aligned}&\Lambda _1(\epsilon ^{\sigma })=\{n\in {{\mathbb {Z}}}^{d}: |\langle n,\alpha \rangle |\geqslant \epsilon ^{\sigma }\},\\&\Lambda _2(\epsilon ^{\sigma })=\{n\in {{\mathbb {Z}}}^{d}: |2\rho -\langle n,\alpha \rangle |\geqslant \epsilon ^{\sigma }\}. \end{aligned}$$
For \(\eta =\epsilon ^{\sigma }\), we define the decomposition \(\mathcal {B}_r=\mathcal {B}_r^{nre}(\epsilon ^{\sigma }) \bigoplus \mathcal {B}_r^{re}(\epsilon ^{\sigma })\) as in (3.1) with A substituted by \(A'\). Recall that su(1, 1) consists of matrices of the form
$$\begin{aligned} \begin{pmatrix} it &{} v\\ \bar{v} &{} -it \end{pmatrix} \end{aligned}$$
with \(t\in {{\mathbb {R}}}\), \(v\in {{\mathbb {C}}}\). Direct computation shows that any \(Y\in \mathcal {B}_r^{nre}(\epsilon ^{\sigma })\) takes the precise form:
$$\begin{aligned} \begin{aligned} Y(\theta )&=\sum _{n\in \Lambda _1(\epsilon ^{\sigma })}\begin{pmatrix} i\hat{t}(n) &{} 0\\ 0 &{} -i\hat{t}(n) \end{pmatrix} e^{i\langle n,\theta \rangle }\\&\quad +\sum _{n\in \Lambda _2(\epsilon ^{\sigma })}\begin{pmatrix} 0 &{} \hat{v}(n)e^{i\langle n,\theta \rangle }\\ \overline{\hat{v}(n)}e^{-i\langle n,\theta \rangle } &{} 0 \end{pmatrix}. \end{aligned} \end{aligned}$$
(5.11)
Since \(\epsilon ^{\sigma }\geqslant 13\Vert A'\Vert ^2 (\epsilon ') ^{\frac{1}{2}}\), we can apply Lemma 3.1 to remove all the non-resonant terms of g, which means there exist \(Y\in \mathcal {B}_r\) and \(g^{re}\in \mathcal {B}_r^{re}(\eta )\) such that
$$\begin{aligned} e^{Y(\theta +\alpha )}(A'e^{g(\theta )})e^{-Y(\theta )}=A'e^{g^{re}(\theta )}, \end{aligned}$$
with \(|Y |_r\leqslant (\epsilon ')^{\frac{1}{2}}\) and \(|g^{re}|_r\leqslant 2\epsilon '\).
Combining with the Diophantine condition on the frequency \(\alpha \) and the Claim 2, we have:
$$\begin{aligned}&\{{{\mathbb {Z}}}^{d}\backslash \Lambda _1(\epsilon ^{\sigma })\}\cap \{n\in {{\mathbb {Z}}}^{d}:|n |\leqslant \kappa ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}\}=\{0\},\\&\{{{\mathbb {Z}}}^{d}\backslash \Lambda _2(\epsilon ^{\sigma })\}\cap \{n\in {{\mathbb {Z}}}^{d}:|n |\leqslant 2^{-\frac{1}{\tau }}\kappa ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}-N\}=\{n_*\}. \end{aligned}$$
Let \(N':=2^{-\frac{1}{\tau }}\kappa ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}-N\), then we can rewrite \(g^{re}(\theta )\) as
$$\begin{aligned} g^{re}(\theta )&=g^{re}_0+g^{re}_1(\theta )+g^{re}_2(\theta )\\&=\begin{pmatrix} i\hat{t}(0) &{}\quad 0 \\ 0 &{}\quad -i\hat{t}(0) \end{pmatrix}+\begin{pmatrix} 0 &{}\quad \hat{v}(n_*)e^{i\langle n_*,\theta \rangle } \\ \overline{\hat{v}(n_*)}e^{-i\langle n_*,\theta \rangle } &{}\quad 0 \end{pmatrix}\\&\quad +\, \sum _{|n |>N'}\hat{g}^{re}(n)e^{i\langle n,\theta \rangle }. \end{aligned}$$
Define the \(4\pi {{\mathbb {Z}}}^d\)-periodic rotation \(Q(\theta )\) as below:
$$\begin{aligned} Q(\theta )=\begin{pmatrix} e^{-\frac{\langle n_*,\theta \rangle }{2}i} &{}\quad 0\\ 0 &{}\quad e^{\frac{\langle n_*,\theta \rangle }{2}i} \end{pmatrix}. \end{aligned}$$
So we have
$$\begin{aligned} |Q(\theta )|_{r'}\leqslant e^{\frac{1}{2}Nr'}\leqslant \epsilon ^{\frac{-r'}{r-r'}}. \end{aligned}$$
(5.12)
One can also show that
$$\begin{aligned} Q(\theta +\alpha )(A'e^{g^{re}(\theta )})Q^{-1}(\theta )=\tilde{A}e^{\tilde{g}(\theta )}, \end{aligned}$$
where
$$\begin{aligned} \tilde{A}=Q(\theta +\alpha )A'Q^{-1}(\theta )=\begin{pmatrix} e^{i(\rho -\frac{\langle n_*,\alpha \rangle }{2})} &{}\quad 0\\ 0 &{}\quad e^{-i(\rho -\frac{\langle n_*,\alpha \rangle }{2})} \end{pmatrix} \end{aligned}$$
(5.13)
and
$$\begin{aligned} \tilde{g}(\theta )=Qg^{re}(\theta )Q^{-1}=Qg^{re}_0Q^{-1}+Qg^{re}_1(\theta )Q^{-1}+Qg^{re}_2(\theta )Q^{-1}. \end{aligned}$$
Moreover,
$$\begin{aligned} Qg^{re}_0Q^{-1}&=g^{re}_0 = \begin{pmatrix} i\hat{t}(0) &{}\quad 0 \\ 0 &{}\quad -i\hat{t}(0) \end{pmatrix} \in su(1,1), \end{aligned}$$
(5.14)
$$\begin{aligned} Qg^{re}_1(\theta )Q^{-1}&=\begin{pmatrix} 0 &{}\quad \hat{v}(n_*) \\ \overline{\hat{v}(n_*)} &{}\quad 0 \end{pmatrix} \in su(1,1). \end{aligned}$$
(5.15)
Now we return back from su(1, 1) to \(sl(2,{{\mathbb {R}}})\). Denote
$$\begin{aligned} L&=M^{-1}(Qg^{re}_0Q^{-1}+Qg^{re}_1(\theta )Q^{-1})M, \end{aligned}$$
(5.16)
$$\begin{aligned} F&=M^{-1}Qg^{re}_2(\theta )Q^{-1}M, \end{aligned}$$
(5.17)
$$\begin{aligned} B&=M^{-1}(Q\circ e^Y \circ P) M, \end{aligned}$$
(5.18)
$$\begin{aligned} \tilde{A}^{'}&=M^{-1}\tilde{A}M, \end{aligned}$$
(5.19)
then we have:
$$\begin{aligned} B(\theta +\alpha )(Ae^{f(\theta )})B^{-1}(\theta )=\tilde{A}^{'}e^{L+F(\theta )}. \end{aligned}$$
(5.20)
By (5.9) and (5.12), we have the following estimates:
$$\begin{aligned} \Vert B\Vert _0\leqslant & {} |e^{Y}|_r \Vert P\Vert \leqslant 8 \left( \frac{\Vert A\Vert }{\kappa }\right) ^{\frac{1}{2}}\left( \frac{2}{r-r'} |\ln \epsilon |\right) ^{\frac{\tau }{2}}, \end{aligned}$$
(5.21)
$$\begin{aligned} |B|_{r'}\leqslant & {} 8\left( \frac{\Vert A\Vert }{\kappa }\right) ^{\frac{1}{2}}\left( \frac{2}{r-r'} |\ln \epsilon |\right) ^{\frac{\tau }{2}}\times \epsilon ^{\frac{-r'}{r-r'}},\end{aligned}$$
(5.22)
$$\begin{aligned} \Vert L \Vert\leqslant & {} \Vert Qg^{re}_0Q^{-1}\Vert + \Vert Qg^{re}_1(\theta )Q^{-1}\Vert \leqslant \epsilon '+\epsilon ' e^{-|n_{*}|r},\end{aligned}$$
(5.23)
$$\begin{aligned} |F|_{r'}\leqslant & {} |Qg^{re}_2(\theta )Q^{-1}|_{r'} \leqslant \frac{2^{4+\tau }\Vert A\Vert |\ln \epsilon |^{\tau }}{\kappa (r-r')^{\tau }}\epsilon e^{-N'(r-r')}(N')^de^{Nr'}. \end{aligned}$$
(5.24)
By (5.23) and (5.24), direct computation shows that
$$\begin{aligned} e^{L+F(\theta )}=e^L+\mathcal {O}(F(\theta ))=e^L(Id+e^{-L}\mathcal {O}(F(\theta )))=e^L e^{f_+{(\theta )}}. \end{aligned}$$
(5.25)
It immediately implies that
$$\begin{aligned} |f_+{(\theta )}|_{r'}\leqslant 2 |F(\theta )|_{r'}\leqslant \frac{2^{5+\tau }\Vert A\Vert |\ln \epsilon |^{\tau }}{\kappa (r-r')^{\tau }}\epsilon e^{-N'(r-r')}(N')^de^{Nr'}\ll \epsilon ^{100}. \end{aligned}$$
Thus, we can rewrite (5.20) as
$$\begin{aligned} B(\theta +\alpha )(Ae^{f(\theta )})B^{-1}(\theta )=A_{+}e^{f_+(\theta )}, \end{aligned}$$
with
$$\begin{aligned} A_+=\tilde{A}^{'}e^L=e^{A''}, \ \ A''\in sl(2,{{\mathbb {R}}}). \end{aligned}$$
(5.26)
Now recall that Baker–Campbell–Hausdorff formula [28] says that
$$\begin{aligned} \ln (e^X e^Y)=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}([X,[X,Y]+[Y,[Y,X]])+\cdots ,\nonumber \\ \end{aligned}$$
(5.27)
where \([X,Y]=XY-YX\) denotes the Lie Bracket and \(\cdots \) denotes the sum of higher order terms. Using this formula and by a simple calculation, (5.26) gives
$$\begin{aligned} MA''M^{-1}=\begin{pmatrix} it &{} v\\ \bar{v} &{} -it \end{pmatrix} \end{aligned}$$
where
$$\begin{aligned} t=\rho -\frac{\langle n_*,\alpha \rangle }{2}+\hat{t}(0)+ \mathrm{higher\, order\, terms} \end{aligned}$$
and
$$\begin{aligned} v=\hat{v}(n_*)+ \mathrm{higher\, order\, terms}. \end{aligned}$$
By (5.8) and (5.23), we obtain \(|t|\leqslant \epsilon ^{\sigma }\) and
$$\begin{aligned} |v |\leqslant \frac{2^{4+\tau }\Vert A\Vert |\ln \epsilon |^{\tau }}{\kappa (r-r')^{\tau }}\epsilon e^{-|n_{*}|r}. \end{aligned}$$
Finally, the following estimate is straightforward:
$$\begin{aligned} \Vert A'' \Vert \leqslant 2(|\rho -\frac{\langle n_*,\alpha \rangle }{2}|+\Vert Qg^{re}_0Q^{-1}\Vert +\Vert Qg^{re}_1(\theta )Q^{-1}\Vert )\leqslant 2\epsilon ^{\sigma }. \end{aligned}$$
(5.28)
This finishes the proof of Theorem 3.1. \(\square \)