Appendix A. Proof of Theorem 3
Note that
$$\begin{aligned}&\mathbf {\pi }\left( \mathbf {B}_{\lambda ,l}(c_k,t)\right) =\mathbf {\pi }\left( \sum _{j=1}^{\infty }(-1)^j\frac{j}{(j+1)!}\mathrm {ad}_{\mathbf {D}_{\lambda ,l-1}(c_k,t)}^j\mathbf {B}_{\lambda ,l-1}(c_k,t)\right) \\&\quad =\sum _{j=1}^{\infty }(-1)^j\frac{j}{(j+1)!}\mathbf {\pi }\left( \mathrm {ad}_{\mathbf {D}_{\lambda ,l-1}(c_k,t)}^j\mathbf {B}_{\lambda ,l-1}(c_k,t)\right) \\&\quad =\left( \sum _{j=1}^{\infty }(-1)^j\frac{j}{(j+1)!}\mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,l-1}(c_k,t)}^j\right) \mathbf {\pi }\left( \mathbf {B}_{\lambda ,l-1}(c_k,t)\right) \\&\quad =-\left( \sum _{j=1}^{\infty }\frac{2j-1}{(2j)!}\mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,l-1}(c_k,t)}^{2j-1}\right) \mathbf {\pi }\left( \mathbf {B}_{\lambda ,l-1}(c_k,t)\right) \\&\quad \quad +\left( \sum _{j=1}^{\infty }\frac{2j}{(2j+1)!}\mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,l-1}(c_k,t)}^{2j}\right) \mathbf {\pi }\left( \mathbf {B}_{\lambda ,l-1}(c_k,t)\right) \\&\quad =-\left( \sum _{j=1}^{\infty }\frac{2j-1}{(2j)!}\rho ^{2j-2}\left( \mathbf {D}_{\lambda ,l-1}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,l-1}(c_k,t)}\mathbf {\pi }\left( \mathbf {B}_{\lambda ,l-1}(c_k,t)\right) \\&\quad \quad +\left( \sum _{j=1}^{\infty }\frac{2j}{(2j+1)!}\rho ^{2j-2}\left( \mathbf {D}_{\lambda ,l-1}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,l-1}(c_k,t)}^2\mathbf {\pi }\left( \mathbf {B}_{\lambda ,l-1}(c_k,t)\right) \end{aligned}$$
where the first equality is due to Definition 2, and the third and last equalities are due to Theorem 2.
Appendix B. Proof of Theorem 4
Recall Definitions 2 and 3 and note that
$$\begin{aligned} \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) =2(t-c_k)\sqrt{\frac{\int _{c_k}^tq(\xi )d\xi }{t-c_k}-\lambda }. \end{aligned}$$
(27)
Note further that, (27) and assumptions (4) and (5) ensure
$$\begin{aligned} \lambda \in \left[ q_{\mathrm {max}}-h_{\mathrm {max}}^{-2},q_{\mathrm {min}}\right]&\Rightarrow \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \in \left[ 0,2h_{\mathrm {max}}\sqrt{q_{\mathrm {max}}-\lambda }\right] \subseteq \left[ 0,2\right] \end{aligned}$$
(28)
$$\begin{aligned} \lambda \in \left[ q_{\mathrm {min}},q_{\mathrm {max}}\right]&\Rightarrow \left| \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right| \le 2h_{\mathrm {max}}\sqrt{q_{\mathrm {max}}-q_{\mathrm {min}}}\le 2\end{aligned}$$
(29)
$$\begin{aligned} \lambda \in \left[ q_{\mathrm {max}},q_{\mathrm {max}}+h_{\mathrm {max}}^{-2}\right]&\Rightarrow \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \in i\left[ 0,2h_{\mathrm {max}}\sqrt{\lambda -q_{\mathrm {min}}}\right] \subseteq i\left[ 0,2\sqrt{2}\right] \end{aligned}$$
(30)
$$\begin{aligned} \lambda \in \left[ q_{\mathrm {max}}+h_{\mathrm {max}}^{-2},+\infty \right)&\Rightarrow \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \in i\left[ 2\left( t-c_k\right) \sqrt{\lambda -q_{\mathrm {max}}},+\infty \right) \end{aligned}$$
(31)
which, together with Definition 4 and Remark 5, lead to the following estimates, in the two uniform regimes (11) and (12):
$$\begin{aligned} \left| \varphi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \right|&\le 2,\text { w.r.t }(11),\end{aligned}$$
(32)
$$\begin{aligned} \left| \phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \right|&\le 1,\text { w.r.t }(11),\end{aligned}$$
(33)
$$\begin{aligned} \left| \phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \rho ^2\left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right|&\le 2,\text { w.r.t }(11),\end{aligned}$$
(34)
$$\begin{aligned} \left| \varphi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \right|&\le \frac{\left( t-c_k\right) ^{-1}}{\sqrt{\lambda -q_{\mathrm {max}}}},\text { w.r.t }(12),\end{aligned}$$
(35)
$$\begin{aligned} \left| \phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \right|&\le \frac{1}{2}\frac{\left( t-c_k\right) ^{-2}}{\lambda -q_{\mathrm {max}}},\text { w.r.t }(12),\end{aligned}$$
(36)
$$\begin{aligned} \left| \phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \rho ^2\left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right|&\le 2,\text { w.r.t }(12). \end{aligned}$$
(37)
1.1 B.1. Estimating \(\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) \ldots \exp \left( \mathbf {D}_{\lambda ,0}(a,c_1)\right) \)
Firstly, in the uniform regime (11), we have
$$\begin{aligned}&e^{\mathbf {D}_{\lambda ,0}(c_k,c_{k+1})} =\cosh \frac{\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }{2} \begin{bmatrix} 1&\quad 0\\ 0&\quad 1 \end{bmatrix} \\&\quad \quad +\frac{\sinh \frac{\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }{2}}{\frac{\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }{2}} \begin{bmatrix} 0&\quad c_{k+1}-c_k\\ \left( \frac{\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }{2}\right) ^2(c_{k+1}-c_k)^{-1}&\quad 0 \end{bmatrix} \\&\quad =\mathcal {O}\left( 1\right) \begin{bmatrix} 1&\quad 0\\ 0&\quad 1 \end{bmatrix} + \mathcal {O}\left( 1\right) \begin{bmatrix} 0&\quad \mathcal {O}\left( 1\right) h_{\mathrm {max}}\\ \mathcal {O}\left( 1\right) h_{\mathrm {min}}^{-1}&\quad 0 \end{bmatrix} \\&\quad =\mathcal {O}\left( 1\right) \begin{bmatrix} 1&\quad 0\\ 0&\quad 1 \end{bmatrix} + \mathcal {O}\left( 1\right) \begin{bmatrix} 0&\quad \mathcal {O}\left( 1\right) h_{\mathrm {max}}\\ \mathcal {O}\left( 1\right) h_{\mathrm {max}}^{-1}&\quad 0 \end{bmatrix} \end{aligned}$$
where we have called upon (6), (28), (29) and (30). Secondly, in the uniform regime (12), we have
$$\begin{aligned}&e^{\mathbf {D}_{\lambda ,0}(c_k,c_{k+1})} =\cos \frac{\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }{2i} \begin{bmatrix} 1&\quad 0\\ 0&\quad 1 \end{bmatrix} \\&\quad \quad +\sin \frac{\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }{2i} \begin{bmatrix} 0&\quad \frac{c_{k+1}-c_k}{(2i)^{-1}\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }\\ -\frac{(2i)^{-1}\rho \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }{c_{k+1}-c_k}&\quad 0 \end{bmatrix} \\&\quad =\mathcal {O}\left( 1\right) \begin{bmatrix} 1&\quad 0\\ 0&\quad 1 \end{bmatrix} +\mathcal {O}\left( 1\right) \begin{bmatrix} 0&\quad \frac{1}{\sqrt{\lambda -\frac{\int _{c_k}^{c_{k+1}}q(\xi )d\xi }{c_{k+1}-c_k}}}\\ -\sqrt{\lambda -\frac{\int _{c_k}^{c_{k+1}}q(\xi )d\xi }{c_{k+1}-c_k}}&\quad 0 \end{bmatrix} \\&\quad =\mathcal {O}\left( 1\right) \begin{bmatrix} 1&\quad 0\\ 0&\quad 1 \end{bmatrix} +\mathcal {O}\left( 1\right) \begin{bmatrix} 0&\quad \mathcal {O}\left( 1\right) \frac{1}{\sqrt{\lambda -q_{\mathrm {max}}}}\\ \mathcal {O}\left( 1\right) \sqrt{\lambda -q_{\mathrm {min}}}&\quad 0 \end{bmatrix} \\&\quad =\mathcal {O}\left( 1\right) \begin{bmatrix} 1&\quad 0\\ 0&\quad 1 \end{bmatrix} +\mathcal {O}\left( 1\right) \begin{bmatrix} 0&\quad \mathcal {O}\left( 1\right) \frac{1}{\sqrt{\lambda -q_{\mathrm {max}}}}\\ \mathcal {O}\left( 1\right) \sqrt{\lambda -q_{\mathrm {max}}}&\quad 0 \end{bmatrix} \end{aligned}$$
where the second equality is due to (27) and (31), and the last equality is due to the fact that (4) ensures that
$$\begin{aligned} \frac{\sqrt{\lambda -q_{\mathrm {min}}}}{\sqrt{\lambda -q_{\mathrm {max}}}} =\sqrt{1+\frac{q_{\mathrm {max}}-q_{\mathrm {min}}}{\lambda -q_{\mathrm {max}}}} \le \sqrt{1+h_{\mathrm {max}}^2(q_{\mathrm {max}}-q_{\mathrm {min}})}\le \sqrt{2}. \end{aligned}$$
1.2 B.2. Estimating \(\mathbf {\pi }\left( \mathbf {B}_{\lambda ,1}(c_k,t)\right) \) and \(\mathbf {\pi }\left( \mathbf {D}_{\lambda ,1}(c_k,t)\right) \)
Finally, we note that (32)–(37), in turn, imply that
$$\begin{aligned}&\varphi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,0}(c_k,t)}\mathbf {\pi }\left( \mathbf {B}_{\lambda ,0}(c_k,t)\right) \\&\quad = \begin{bmatrix} \varphi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \frac{q(t)-\frac{\int _{c_k}^t q(\xi )d\xi }{t-c_k}}{t-c_k}(t-c_k)^2\\ 0\\ 0 \end{bmatrix}\\&\quad = {\left\{ \begin{array}{ll} \delta _{\left| q'\right| } \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}^2\right) \\ 0\\ 0 \end{bmatrix} ,&{}\text { w.r.t }(11),\\ \delta _{\left| q'\right| } \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}\right) (\lambda -q_{\mathrm {max}})^{-\frac{1}{2}}\\ 0\\ 0 \end{bmatrix} ,&{}\text { w.r.t }(12), \end{array}\right. } \end{aligned}$$
and
$$\begin{aligned}&\phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,0}(c_k,t)}^2\mathbf {\pi }\left( \mathbf {B}_{\lambda ,0}(c_k,t)\right) \\&\quad = \begin{bmatrix} 0\\ -2\phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \frac{q(t)-\frac{\int _{c_k}^t q(\xi )d\xi }{t-c_k}}{t-c_k}(t-c_k)^3\\ \frac{1}{2}\phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \rho ^2\left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \frac{q(t)-\frac{\int _{c_k}^t q(\xi )d\xi }{t-c_k}}{t-c_k}(t-c_k) \end{bmatrix}\\&\quad = {\left\{ \begin{array}{ll} \delta _{\left| q'\right| } \begin{bmatrix} 0\\ \mathcal {O}\left( h_{\mathrm {max}}^3\right) \\ \mathcal {O}\left( h_{\mathrm {max}}\right) \end{bmatrix} ,&{}\text { w.r.t }(11),\\ \delta _{\left| q'\right| } \begin{bmatrix} 0\\ \mathcal {O}\left( h_{\mathrm {max}}\right) (\lambda -q_{\mathrm {max}})^{-1}\\ \mathcal {O}\left( h_{\mathrm {max}}\right) \end{bmatrix} ,&{}\text { w.r.t }(12), \end{array}\right. } \end{aligned}$$
which, according to Theorem 3, lead to
$$\begin{aligned} \mathbf {\pi }\left( \mathbf {B}_{\lambda ,1}(c_k,t)\right)&=\varphi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,0}(c_k,t)}\mathbf {\pi }\left( \mathbf {B}_{\lambda ,0}(c_k,t)\right) \nonumber \\&\quad +\phi \left( \rho \left( \mathbf {D}_{\lambda ,0}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,0}(c_k,t)}^2\mathbf {\pi }\left( \mathbf {B}_{\lambda ,0}(c_k,t)\right) \nonumber \\&= {\left\{ \begin{array}{ll} \delta _{\left| q'\right| } \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}^2\right) \\ \mathcal {O}\left( h_{\mathrm {max}}^3\right) \\ \mathcal {O}\left( h_{\mathrm {max}}\right) \end{bmatrix} ,&{}\text { w.r.t }(11),\\ \delta _{\left| q'\right| } \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}\right) (\lambda -q_{\mathrm {max}})^{-\frac{1}{2}}\\ \mathcal {O}\left( h_{\mathrm {max}}\right) (\lambda -q_{\mathrm {max}})^{-1}\\ \mathcal {O}\left( h_{\mathrm {max}}\right) \end{bmatrix} ,&{}\text { w.r.t }(12). \end{array}\right. } \end{aligned}$$
1.3 B.3. Estimating \(\mathbf {\pi }\left( \mathbf {B}_{\lambda ,l}(c_k,t)\right) \) and \(\mathbf {\pi }\left( \mathbf {D}_{\lambda ,l}(c_k,t)\right) \) for \(l\ge 2\)
Follows by induction.
1.3.1 B.3.1. First step: \(l=2\)
Given Definition 4 and the uniform estimates for \(\mathbf {\pi }\left( \mathbf {B}_{\lambda ,1}(c_k,t)\right) \) in the previous subsection, it is now clear that
$$\begin{aligned} \varphi \left( \rho \left( \mathbf {D}_{\lambda ,1}(c_k,t)\right) \right)&= {\left\{ \begin{array}{ll} -\frac{1}{2}+\delta _{\left| q'\right| }^2\mathcal {O}\left( h_{\mathrm {max}}^6\right) ,&{}\text { w.r.t }(11),\\ -\frac{1}{2}+\delta _{\left| q'\right| }^2\mathcal {O}\left( h_{\mathrm {max}}^{4}\right) \left( \lambda -q_{\mathrm {max}}\right) ^{-1},&{}\text { w.r.t }(12), \end{array}\right. } \\ \phi \left( \rho \left( \mathbf {D}_{\lambda ,1}(c_k,t)\right) \right)&= {\left\{ \begin{array}{ll} \frac{1}{3}+\delta _{\left| q'\right| }^2\mathcal {O}\left( h_{\mathrm {max}}^6\right) ,&{}\text { w.r.t }(11),\\ \frac{1}{3}+\delta _{\left| q'\right| }^2\mathcal {O}\left( h_{\mathrm {max}}^{4}\right) \left( \lambda -q_{\mathrm {max}}\right) ^{-1},&{}\text { w.r.t }(12), \end{array}\right. } \end{aligned}$$
and, according to Theorem 3, that
$$\begin{aligned} \mathbf {\pi }\left( \mathbf {B}_{\lambda ,2}(c_k,t)\right)&=\varphi \left( \rho \left( \mathbf {D}_{\lambda ,1}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,1}(c_k,t)}\mathbf {\pi }\left( \mathbf {B}_{\lambda ,1}(c_k,t)\right) \\&\quad +\phi \left( \rho \left( \mathbf {D}_{\lambda ,1}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,1}(c_k,t)}^2\mathbf {\pi }\left( \mathbf {B}_{\lambda ,1}(c_k,t)\right) \\&= {\left\{ \begin{array}{ll} \delta _{\left| q'\right| }^2 \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}^5\right) \\ \mathcal {O}\left( h_{\mathrm {max}}^6\right) \\ \mathcal {O}\left( h_{\mathrm {max}}^4\right) \end{bmatrix} ,&{}\text { w.r.t }(11),\\ \delta _{\left| q'\right| }^2 \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}^{3}\right) (\lambda -q_{\mathrm {max}})^{-1}\\ \mathcal {O}\left( h_{\mathrm {max}}^{3}\right) (\lambda -q_{\mathrm {max}})^{-\frac{3}{2}}\\ \mathcal {O}\left( h_{\mathrm {max}}^{3}\right) (\lambda -q_{\mathrm {max}})^{-\frac{1}{2}} \end{bmatrix} ,&{}\text { w.r.t }(12). \end{array}\right. } \end{aligned}$$
1.3.2 B.3.2. Induction step: \(l\Rightarrow l+1\)
Given the induction claim, it is now clear that
$$\begin{aligned} \varphi \left( \rho \left( \mathbf {D}_{\lambda ,l}(c_k,t)\right) \right)&= {\left\{ \begin{array}{ll} -\frac{1}{2}+\delta _{\left| q'\right| }^{2^l}\mathcal {O}\left( h_{\mathrm {max}}^{3\times 2^l}\right) ,&{}\text { w.r.t }(11),\\ -\frac{1}{2}+\delta _{\left| q'\right| }^{2^l}\mathcal {O}\left( h_{\mathrm {max}}^{2^{l+1}}\right) (\lambda -q_{\mathrm {max}})^{-2^{l-1}},&{}\text { w.r.t }(12) \end{array}\right. } \\ \phi \left( \rho \left( \mathbf {D}_{\lambda ,l}(c_k,t)\right) \right)&= {\left\{ \begin{array}{ll} \frac{1}{3}+\delta _{\left| q'\right| }^{2^l}\mathcal {O}\left( h_{\mathrm {max}}^{3\times 2^l}\right) ,&{}\text { w.r.t }(11),\\ \frac{1}{3}+\delta _{\left| q'\right| }^{2^l}\mathcal {O}\left( h_{\mathrm {max}}^{2^{l+1}}\right) (\lambda -q_{\mathrm {max}})^{-2^{l-1}},&{}\text { w.r.t }(12), \end{array}\right. } \end{aligned}$$
and, according to Theorem 3, that
$$\begin{aligned} \mathbf {\pi }\left( \mathbf {B}_{\lambda ,l+1}(c_k,t)\right)&=\varphi \left( \rho \left( \mathbf {D}_{\lambda ,l}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,l}(c_k,t)}\mathbf {\pi }\left( \mathbf {B}_{\lambda ,l}(c_k,t)\right) \\&\quad +\phi \left( \rho \left( \mathbf {D}_{\lambda ,l}(c_k,t)\right) \right) \mathbf {\fancyscript{C}}_{\mathbf {D}_{\lambda ,l}(c_k,t)}^2\mathbf {\pi }\left( \mathbf {B}_{\lambda ,l}(c_k,t)\right) \\&= {\left\{ \begin{array}{ll} \delta _{\left| q'\right| }^{2^l} \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}^{3\times 2^{l}-1}\right) \\ \mathcal {O}\left( h_{\mathrm {max}}^{3\times 2^{l}}\right) \\ \mathcal {O}\left( h_{\mathrm {max}}^{3\times 2^{l}-2}\right) \end{bmatrix} ,&{}\text { w.r.t }(11),\\ \delta _{\left| q'\right| }^{2^l} \begin{bmatrix} \mathcal {O}\left( h_{\mathrm {max}}^{2^{l+1}-1}\right) (\lambda -q_{\mathrm {max}})^{-\frac{2^{l}}{2}}\\ \mathcal {O}\left( h_{\mathrm {max}}^{2^{l+1}-1}\right) (\lambda -q_{\mathrm {max}})^{-\frac{2^{l}+1}{2}}\\ \mathcal {O}\left( h_{\mathrm {max}}^{2^{l+1}-1}\right) (\lambda -q_{\mathrm {max}})^{-\frac{2^{l}-1}{2}} \end{bmatrix} ,&{}\text { w.r.t }(12). \end{array}\right. } \end{aligned}$$
Appendix C. Proof of Theorem 5
Recall the Baker–Campbell–Hausdorff-type formulas
$$\begin{aligned} e^{\mathbf {X}}e^{\mathbf {Y}}&=e^{\mathbf {X}+\mathbf {Y}+\frac{1}{2}\left[ \mathbf {X},\mathbf {Y}\right] +\frac{1}{12}\left( \left[ \mathbf {X},\left[ \mathbf {X},\mathbf {Y}\right] \right] +\left[ \mathbf {Y},\left[ \mathbf {Y},\mathbf {X}\right] \right] \right) +\cdots }\end{aligned}$$
(38)
$$\begin{aligned} e^{\mathbf {X}}e^{\mathbf {Y}}e^{-\mathbf {X}}&=e^{\mathbf {Y}+\left[ \mathbf {X},\mathbf {Y}\right] +\frac{1}{2}\left[ \mathbf {X},\left[ \mathbf {X},\mathbf {Y}\right] \right] +\frac{1}{6}\left[ \mathbf {X},\left[ \mathbf {X},\left[ \mathbf {X},\mathbf {Y}\right] \right] \right] +\cdots }\end{aligned}$$
(39)
$$\begin{aligned}&=\exp \left( \mathrm {Ad}_{\exp \left( \mathbf {X}\right) }\left( \mathbf {Y}\right) \right) . \end{aligned}$$
(40)
The local error can be written as
$$\begin{aligned}&\mathbf {L}_{\lambda ,n}(c_k,c_{k+1})\nonumber \\&\quad =\log \left( \mathbf {F}_{\lambda }(c_k,c_{k+1})\tilde{\mathbf {F}}_{\lambda ,n}^{-1}(c_k,c_{k+1})\right) \nonumber \\&\quad =\log \left( \left( \prod _{l=0}^{\infty }e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) \left( \prod _{l=0}^{n}e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) ^{-1}\right) \nonumber \\&\quad =\log \left( \left( \prod _{l=0}^{n}e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) \left( \prod _{l=n+1}^{\infty }e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) \left( \prod _{l=0}^{n}e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) ^{-1}\right) \nonumber \\&\quad =\log \left( \left( \prod _{l=0}^{n}e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) e^{\mathbf {D}_{\lambda ,n+1}(c_k,c_{k+1})+\text {h.o.t.}}\left( \prod _{l=0}^{n}e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) ^{-1}\right) \nonumber \\&\quad =\log \left( e^{\mathbf {D}_{\lambda ,0}(c_k,c_{k+1})}e^{\mathbf {D}_{\lambda ,n+1}(c_k,c_{k+1})+\text {h.o.t.}}e^{-\mathbf {D}_{\lambda ,0}(c_k,c_{k+1})}\right) \nonumber \\&\quad =\mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }\left( \mathbf {D}_{\lambda ,n+1}(c_k,c_{k+1})+\text {h.o.t.}\right) \nonumber \\&\quad =\mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }\left( \mathbf {D}_{\lambda ,n+1}(c_k,c_{k+1})\right) +\text {h.o.t.} \end{aligned}$$
(41)
where the first and second equalities are due to Definition 6, the fourth equality is due to (38), the fifth equality is due to (39), and the sixth equality is due to (40). The local error expression (41), together with Theorem 4, yields the desired estimate.
The global error obeys the recursion relation with initial condition
$$\begin{aligned} \mathbf {G}_{\lambda ,n}(c_1)=\mathbf {L}_{\lambda ,n}(a,c_1) \end{aligned}$$
(42)
and general rule
$$\begin{aligned}&\mathbf {G}_{\lambda ,n}(c_{k+1})\nonumber \\&\quad =\log \left( \mathbf {Y}_{\lambda }(c_{k+1})\tilde{\mathbf {Y}}_{\lambda ,n}^{-1}(c_{k+1})\right) \nonumber \\&\quad =\log \left( \mathbf {F}_{\lambda }(c_k,c_{k+1})\mathbf {Y}_{\lambda }(c_k)\tilde{\mathbf {Y}}_{\lambda ,n}^{-1}(c_k)\tilde{\mathbf {F}}_{\lambda ,n}^{-1}(c_k,c_{k+1})\right) \nonumber \\&\quad =\log \left( \mathbf {F}_{\lambda }(c_k,c_{k+1})e^{\mathbf {G}_{\lambda ,n}(c_k)}\tilde{\mathbf {F}}_{\lambda ,n}^{-1}(c_k,c_{k+1})\right) \nonumber \\&\quad =\log \left( e^{\mathbf {L}_{\lambda ,n}(c_k,c_{k+1})}\tilde{\mathbf {F}}_{\lambda ,n}(c_k,c_{k+1})e^{\mathbf {G}_{\lambda ,n}(c_k)}\tilde{\mathbf {F}}_{\lambda ,n}^{-1}(c_k,c_{k+1})\right) \nonumber \\&\quad =\log \left( e^{\mathbf {L}_{\lambda ,n}(c_k,c_{k+1})}\left( \prod _{l=0}^{n}e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) e^{\mathbf {G}_{\lambda ,n}(c_k)}\left( \prod _{l=0}^{n}e^{\mathbf {D}_{\lambda ,l}(c_k,c_{k+1})}\right) ^{-1}\right) \nonumber \\&\quad =\log \left( e^{\mathbf {L}_{\lambda ,n}(c_k,c_{k+1})}e^{\mathbf {D}_{\lambda ,0}(c_k,c_{k+1})}e^{\mathbf {G}_{\lambda ,n}(c_k)+\text {h.o.t.}}e^{-\mathbf {D}_{\lambda ,0}(c_k,c_{k+1})}\right) \nonumber \\&\quad =\log \left( e^{\mathbf {L}_{\lambda ,n}(c_k,c_{k+1})}\exp \left( \mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }\left( \mathbf {G}_{\lambda ,n}(c_k)+\text {h.o.t.}\right) \right) \right) \nonumber \\&\quad =\log \left( e^{\mathbf {L}_{\lambda ,n}(c_k,c_{k+1})}\exp \left( \mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }\left( \mathbf {G}_{\lambda ,n}(c_k)\right) +\text {h.o.t.}\right) \right) \nonumber \\&\quad =\mathbf {L}_{\lambda ,n}(c_k,c_{k+1})+\mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }\left( \mathbf {G}_{\lambda ,n}(c_k)\right) +\text {h.o.t.} \end{aligned}$$
(43)
where the first, second, third, fourth and fifth equalities are due to Definition 6, the sixth equality is due to (39), the seventh equality is due to (40), and the last equality is due to (38). The global error expressions (42) and (43) lead to
$$\begin{aligned} \mathbf {G}_{\lambda ,n}(c_{k+1})&=\mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) }\left( \mathbf {D}_{\lambda ,n+1}(c_k,c_{k+1})\right) \\&\quad +\mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) \exp \left( \mathbf {D}_{\lambda ,0}(c_{k-1},c_k)\right) }\left( \mathbf {D}_{\lambda ,n+1}(c_{k-1},c_k)\right) \\&\quad +\cdots +\\&\quad +\mathrm {Ad}_{\exp \left( \mathbf {D}_{\lambda ,0}(c_k,c_{k+1})\right) \cdots \exp \left( \mathbf {D}_{\lambda ,0}(a,c_1)\right) }\left( \mathbf {D}_{\lambda ,n+1}(a,c_1)\right) \\&\quad +\text {h.o.t.} \end{aligned}$$
which, together with (6) and Theorem 4, result in the desired estimate.
