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\begin{equation}\tag{BO} \tfrac{d}{dt} q = \h q'' - 2qq' , \end{equation}

\begin{equation} Q_c(t,x) = \frac{2c}{c^2 (x-ct)^2 + 1} \qtq{with} c>0. \end{equation}

\begin{align} q(t,x) \mapsto q_\lambda(t,x) = \lambda q(\lambda^2 t, \lambda x) \qquad\text{for $\lambda>0$} \end{align}

\begin{equation} P(q) = \int \tfrac{1}{2}q^2\, dx \qtq{and} \hbo(q) = \int {\tfrac{1}{2} q\h q' } - \tfrac{1}{3} q^3 \, dx, \end{equation}

\begin{equation} \{ F,G \} = \int \tfrac{\del F}{\del q}(x) \cdot \big( \tfrac{\del G}{\del q} \big)'(x)\, dx , \end{equation}

\begin{equation} q(t,x) = \widetilde q(t,x-2ct) + c. \end{equation}

\begin{equation} H_2 := \int \tfrac12 \big[q'\bigr]^2 - \tfrac34 q^2 \h q' + \tfrac14 q^4 \,dx . \end{equation}

\begin{equation} \mc L f = -i f' - C_+\bigl( q f\bigr) \end{equation}

\begin{equation} \mc P := - i\partial^2 - 2 \partial C_+q + 2 q'_+ \qtq{or} \mc P - i\mc{L}^2 = i C_+(\h q') - iC_+qC_+q . \end{equation}

\begin{equation} -i \partial_x \olN - C_+( q \olN) = z \olN - z \quad\text{with $\olN(x)\to 1$ as $x\to+\infty$} \end{equation}

\begin{equation} W = 1 + (\mc L_0 - z)^{-1} C_+(q W), \qtq{where} z\in\C\setminus[0,\infty) \end{equation}

\begin{equation} -i m' - C_+[q(m+1)] +\kappa m =0 \qtq{or equivalently,} m = (\mc L + \kappa)^{-1} C_+ q . \end{equation}

\begin{equation} \beta(\kappa;q) := \int q(x) m(x;\kappa,q)\, dx = \langle q_+, (\mc L +\kappa)^{-1} q_+ \rangle_{L^2_+} . \end{equation}

\begin{equation} \beta(\kappa;q) = \kappa^{-1} P(q) - \kappa^{-2} \hbo(q) + \kappa^{-3} H_2(q) + \bigO(\kappa^{-4}). \end{equation}

\begin{align} e^{tJ\nabla \hbo}(q^0) - e^{tJ\nabla \hbo}(\tilde q^{\:\!0}) &= e^{tJ\nabla \hbo}(q^0) - e^{tJ\nabla H_\kappa}(q^0) \notag \\ &+ e^{tJ\nabla H_\kappa }(q^0) - e^{tJ\nabla H_\kappa}(\tilde q^{\:\!0}) \\ &\qquad+ e^{tJ\nabla H_\kappa}(\tilde q^{\:\!0}) - e^{tJ\nabla \hbo}(\tilde q^{\:\!0}). \notag \end{align}

\begin{align} e^{tJ\nabla \hbo}(q^0) - e^{tJ\nabla H_\kappa}(q^0) = \bigl[ e^{tJ\nabla(\hbo - H_\kappa) } - \Id\bigr]\circ e^{tJ\nabla H_\kappa}(q^0) . \end{align}

\begin{equation} \hk (q):= \begin{cases} \kappa P(q) - \kappa^2 \beta(\kappa;q) & \text{on $\R$,} \\ \bigl[ \kappa + \tint \bigr] P(q) - \kappa^2 \beta(\kappa;q) + \tfrac{\kappa}2 \btint^2 + \tfrac16 \btint^3 & \text{on $\T$.} \end{cases} \end{equation}

\begin{equation} 2q= \tfrac{1}{w+\kappa} \h (w') + \h \bigl[\tfrac{w'}{w+\kappa}\bigr] +\tfrac{2\kappa w}{w+\kappa}. \end{equation}

\begin{equation} \tfrac{d}{dt} w = \h w'' - 2qw', \end{equation}

\begin{align} q(\vec t;q_0) = \Bigl[\exp\bigl\{ {\textstyle\sum} t_i J\nabla H_i \bigr\}q_0 \Bigr] ( x=0 ) \end{align}

\begin{equation} H_\phi(q) := \langle q_+, \phi(\mc L)q_+\rangle \end{equation}

\begin{equation} q(\phi;q_0) = \bigl( e^{ J\nabla H_\phi} q_0\bigr)( x=0 ). \end{equation}

\begin{equation} C_+\bigl( e^{ J\nabla H_\phi} q_0\bigr)( x+iy ) = \tfrac{1}{2\pi i} I_+ \Big( \big( X - t \psi(\mc L_{q_0}) - x - iy \big)^{-1} q^0_+\Big) \end{equation}

\begin{equation} \psi(E) = \phi(E) + E\phi'(E). \end{equation}

\begin{align*} \hat f(\xi) = \tfrac{1}{\sqrt{2\pi}} \int_\R e^{-i\xi x} f(x)\,dx \qtq{so} f(x) = \tfrac{1}{\sqrt{2\pi}} \int_\R e^{i\xi x} \hat f(\xi)\,d\xi \end{align*}

\begin{align*} \hat f(\xi) = \int_0^1 e^{- i\xi x} f(x)\,dx \qtq{so} f(x) = \sum_{\xi\in 2\pi\Z} \hat f(\xi) e^{i\xi x}. \end{align*}

\begin{align*} \|f\|_{L^2(\mathbb R)}=\|\hat f\|_{L^2(\mathbb R)} \qtq{and} \|f\|_{L^2(\mathbb T)}=\sum_{\xi\in 2\pi \mathbb Z}|\hat f(\xi)|^2. \end{align*}

\begin{equation} \wh{\h f}(\xi) = -i\sgn(\xi)\wh{f}(\xi) \end{equation}

\begin{align*} %\end{align*}

\begin{equation} \wh{C_\pm f}(\xi) = 1_{[0,\infty)}(\pm\xi) \wh{f}(\xi) \end{equation}

\begin{align} C_++C_-=1 \text{only on the line; on the circle,} C_+ f + C_- f = f + \textstyle \int \!f. \end{align}

\begin{align} s\in (-\tfrac12,0) \qtq{and define} \eps:= \tfrac12( \tfrac 12-|s|)\in (0,\tfrac14). \end{align}

\begin{align*} \norm{fg}_{H^{2r-1}}^2 &= \frac{1}{2\pi} \int_0^\infty \frac{1}{(\xi+1)^{4|r|+2}} \bigg| \int_0^\xi \wh{f}(\xi-\eta) \wh{g}(\eta)\,d\eta\bigg|^2\,d\xi. \end{align*}

\[ \limsup_{\delta\to0} \ \sup_{q\in Q} \ \sup_{|y|<\delta}\|q(\cdot +y) - q(\cdot)\|_{H^\sigma} = 0. \]

\begin{equation} \lim_{ \kappa\to\infty}\sup_{q\in Q}\int_{|\xi|\geq \kappa} |\wh{q}(\xi)|^2 (|\xi|+1)^{2\sigma}\, d\xi =0\quad\text{on $\R$} \end{equation}

\begin{equation} \lim_{ \kappa\to\infty}\sup_{q\in Q}\sum_{|\xi|\geq \kappa} |\wh{q}(\xi)|^2 (|\xi|+1)^{2\sigma}=0\quad\text{on $\T$}. \end{equation}

\begin{equation} \begin{aligned} \norm{ C_+qR_0(\kappa)C_+f }_{H^s} &\lesssim \kappa^{-2\eps} \norm{q}_{H^s} \norm{f_+}_{H^s_\kappa} ,\\ \norm{ C_+qR_0(\kappa)C_+f }_{H^s_\kappa} &\lesssim \kappa^{-2\eps} \norm{q}_{H^s_\kappa} \norm{f_+}_{H^s_\kappa} , \end{aligned} \end{equation}

\begin{align*} \norm{ C_+qR_0(\kappa)C_+f }_{H^s} ^2 &= \frac{1}{2\pi} \int_0^\infty \frac{1}{(\xi+1)^{2|s|}} \bigg| \int_0^\infty \wh{q}(\xi-\eta) \frac{\wh{f}(\eta)}{\eta+\kappa}\,d\eta \bigg|^2 d\xi . \end{align*}

\begin{align} \bigl| \langle f, qf\rangle \bigr| \lesssim \kappa^{-2\eps} \norm{q}_{H^s_\kappa} \|f\|_{H^{1/2}_\kappa}^2 = \kappa^{-2\eps} \norm{q}_{H^s} \langle f, (\mc L_0 + \kappa) f \rangle. \end{align}

\begin{equation} R(\kappa;q) = (\mc{L}+\kappa)^{-1} = R_0(\kappa)\sum_{\ell\geq 0}\bigl[C_+qR_0(\kappa)\bigr]^\ell, \end{equation}

\begin{equation} R(\kappa)-R_0(\kappa) = R_0(\kappa) C_+ q \sqrt{R_0(\kappa)} \cdot \sqrt{R_0(\kappa)} \bigl[ 1 + C_+ q R(\kappa) \bigr]. \end{equation}

\begin{equation} q\in \BA:=\{ \text{real-valued }q\in H^s : \norm{q}_{H^s} \leq A \} \qtq{and} \kappa \geq C_s \bigl( 1 + A \bigr)^{\frac1{2\eps}}. \end{equation}

\begin{equation} C_+\big( f\,\ol{\mc{L}g} - \ol{g}\,\mc{L}f \big) = iC_+\big(f\ol{g}\big)' + f[1-C_-](q_+\ol{g}) . \end{equation}

\begin{align*} C_+\big( f\,\ol{\mc{L}g} - \ol{g}\,\mc{L}f \big) &= C_+\big\{ if\ol{g}' - fC_-(q\ol{g}) + if'\ol{g} + \ol{g}C_+(qf) \big\} \\ &= iC_+\big(f\ol{g}\big)' + C_+\bigl\{ \ol{g}qf - fC_-(q\ol{g}) \big\}\\ &= iC_+\big(f\ol{g}\big)' + C_+\bigl\{ f [1-C_-] (q\ol{g}) \big\}\\ &= iC_+\big(f\ol{g}\big)' + f [1-C_-] (q\ol{g}). \end{align*}

\begin{align} \lim_{y\to\infty} \pi y f(iy) = \lim_{y\to\infty} \int \frac{y^2}{x^2+y^2} f(x) \,dx = \lim_{\xi\downarrow 0} \tfrac{\sqrt{2\pi}}{2}\widehat f(\xi) \end{align}

\begin{align} I_+(f) := \lim_{y\to\infty} 2\pi y f(iy) = \lim_{y\to\infty} \bigl\langle \chi_y, f \bigr\rangle = \lim_{\xi\downarrow 0} \sqrt{{2\pi}} \widehat f(\xi) \ \text{with}\ \chi_y(x) = \tfrac{iy}{x+iy}. \end{align}

\begin{align} f(z) = \tfrac{1}{2\pi i} I_+\bigl( (X-z)^{-1} f \bigr) = \lim_{y\to\infty} \tfrac{1}{2\pi i} \bigl\langle \chi_y, (X-z)^{-1} f \bigr\rangle \end{align}

\begin{equation} [X,C_+q] f = \tfrac{i}{2\pi} q_+ I_+(f), \qtq{and} [X,\mc L ] f = i - \tfrac{i}{2\pi} q_+ I_+(f) . \end{equation}

\begin{equation} m' = -i\kappa m + i C_+[q(m+1)], \end{equation}

\begin{equation} m(x;\kappa,q) := R(\kappa,q) q_+ = R_0(\kappa) \sum_{\ell\geq 1} [C_+q R_0(\kappa)]^{\ell-1} q_+ \end{equation}

\begin{equation} m(x+h;\kappa,q) = m(x;\kappa,q(\cdot+h)) \quad\text{for all }h\in\R . \end{equation}

\begin{equation} \snorm{ m^{(\sigma)} }_{H^{s+1}_\kappa} \lesssim \sum_{\ell=1}^\infty \ell^\sigma 2^{\sigma-\ell} \Bigl( 1 + \| q\|_{H^{s}_\kappa}\Bigr)^\sigma \| q \|_{H^{s+\sigma}_\kappa} < \infty \end{equation}

\begin{equation} dm|_q(g)= \frac{d}{d\theta} m(x;\kappa,q+\theta g) \bigg|_{\theta=0}= R(\kappa,q) \bigl[ (m+1)C_+g\bigr], \end{equation}

\begin{equation} dm|_0(g)= R_0(\kappa)C_+ g . \end{equation}

\begin{equation} \beta(\kappa;q): = \int q(x) m(x;\kappa,q)\,dx = \int q(x) \ol{m}(x;\kappa,q)\,dx = \bigl\langle q_+, (\mc L+\kappa)^{-1} q_+ \bigr\rangle \end{equation}

\begin{equation} \tfrac{\del\beta}{\del q} = m + \ol m + |m|^2 \end{equation}

\begin{align*} \int q(x) m(x;\kappa,q)\,dx = \langle q, m\rangle = \langle q_+, m\rangle = \bigl\langle q_+, (\mc L+\kappa)^{-1} q_+ \bigr\rangle. \end{align*}

\begin{align*} d\beta|_q(f) &= \int f m + q \cdot R(\kappa,q)[(m+1)f_+] \,dx \\ &= \int f m + \ol{R(\kappa,q)q_+} \cdot (m+1)f \,dx\\ &= \int [m +\ol{m}(m+1) ] f \,dx, \end{align*}

\begin{align} \int_{\kappa}^\infty \!\vk^{2s} \langle q_+, R_0(\vk)q_+\rangle \,d\vk &= \int_{0}^\infty \! \int_{\kappa}^\infty \! \vk^{2s} \frac{|\wh{q}(\xi)|^2}{\xi+\vk}\,d\vk\,d\xi \simeq_s \norm{q_+}_{H^{s}_\kappa}^2. \end{align}

\begin{align*} Q_{**}= \Bigl\{ q(b) \Big|\, q:[a,b]\to H^s \text{ is continuous, } q(a)\in Q, \text{ and } \beta(z;q(t))\equiv \beta(z;q(a))\Bigr\}, \end{align*}

\begin{equation} \kappa = C_s \bigl( 1 + 2 C_sA \bigr)^{\frac1{2\eps}}. \end{equation}

\begin{equation} m := m(x;\kappa,q) \qtq{and} n := m(x;\vk,q) \end{equation}

\begin{equation} \lim_{ \kappa\to\infty}\ \sup_{q\in Q}\norm{m}_{H^{s+1}_\kappa} = 0 \qtq{and} \lim_{ \kappa\to\infty}\ \sup_{q\in Q}\norm{ \mc{L} R(\kappa,q) n}_{H^{s+1}} =0 \end{equation}

\begin{align*} \norm{ \mc{L} R(\kappa,q) n}_{H^{s+1}} = \norm{ R(\vk,q) \mc{L} m}_{H^{s+1}} &\lesssim \norm{ \mc{L} m }_{H^{s}}\lesssim (1+\|q\|_{H^s})\norm{ m}_{H^{s+1}} . \end{align*}

\begin{align} \tfrac{d}{dt} q_+ &= \mc{P} q_+ = - i q_+'' - 2C_+(qq_+)' + 2q_+ q'_+ \\ \tfrac{d}{dt} m &= - im'' - 2C_+([q-q_+] m)' - 2q_+m' \\ \tfrac{d}{dt} \beta(\kappa) &= 0 . \end{align}

\begin{align} \mc{P}q_+ = - iq''_+ - 2C_+(qq_+)' + 2q_+'q_+= C_+\bigl(\h q'' - 2qq'\bigr) = \tfrac{d}{dt} q_+\,. \end{align}

\begin{align*} \tfrac{d}{dt} m &= [\mc{P}, R(\kappa) ] q_+ + R(\kappa) \mc P q_+ = \mc P m . \end{align*}

\begin{align*} \tfrac{d}{dt} \beta(\kappa) &= \bigl\langle \mc P q_+, R(\kappa) q_+\bigr\rangle + \bigl\langle q_+, [\mc P, R(\kappa) \:\!] q_+\bigr\rangle + \bigl\langle q_+, R(\kappa) \mc P q_+\bigr\rangle \\ &=\bigl\langle \mc P q_+, R(\kappa) q_+\bigr\rangle +\bigl\langle q_+, \mc P R(\kappa) q_+\bigr\rangle. \end{align*}

\begin{align} \int \ol{m} n \,dx = \int m \ol{n} \,dx = \bigl\langle (\mc L+\vk)^{-1} q_+, (\mc L+\kappa)^{-1} q_+ \bigr\rangle = - \frac{\beta(\kappa)-\beta(\vk)}{\kappa-\vk} \end{align}

\begin{align} \kappa \int \ol{m} \,dx = \kappa \int m \,dx = \int qm \,dx + \int q \,dx = \beta(\kappa) + \int q \,dx \end{align}

\begin{align} \langle 1, (\mc L + \kappa)^{-1} 1\rangle = \kappa^{-1} + \kappa^{-2} \beta(\kappa;q) + \kappa^{-2}\int q\,dx . \end{align}

\begin{align*} (\mc L + \kappa)^{-1} 1 = (\mc L_0 + \kappa)^{-1} 1 + (\mc L + \kappa)^{-1}C_+ q(\mc L_0 + \kappa)^{-1} 1 = \kappa^{-1} (1 + m) . \end{align*}

\begin{equation} \kappa \int m+ \ol{m} +|m|^2 \,dx = 2\beta(\kappa) - \kappa\frac{\partial\beta}{\partial\kappa}+ 2\int q \,dx, \end{equation}

\begin{equation} \begin{aligned} \h (m\ol{n}+m+\ol{n})' + i(m+1)\ol{n}' -im'(\ol{n}+1) - 2q (m+1)(\ol{n}+1)& \\ {}+ i(\kappa - \vk)\h (m\ol{n}+m+\ol{n}) + (\kappa + \vk)(m\ol{n}+m+\ol{n}) &= 0 , \end{aligned} \end{equation}

\begin{equation} \text{\upshape LHS\eqref{m ODE 1}} = \vk \int m \,dx + \kappa\int \ol n \,dx . \end{equation}

\begin{align*} \text{LHS}\eqref{m ODE 1} ={} &\kappa ( 1+i\h )\ol{n} + \vk ( 1 -i\h )m + (1+i\h) \big[ (\ol{n}+1)C_+(q(m+1)) \big] \\ &+ (1-i\h) \big[ (m+1)C_-(q(n+1)) \big] - 2q(m+1)(\ol{n}+1) . \end{align*}

\begin{align*} \text{LHS}\eqref{m ODE 1} ={} &\kappa ( 1+i\h )\ol{n} + \vk ( 1 -i\h )m + (1+i\h) \big[ (\ol{n}+1)[C_+-1] (q(m+1)) \big] \\ &+ (1-i\h) \big[ (m+1)[C_- -1] (q(n+1)) \big] . \end{align*}

\begin{equation} \{ \beta(\kappa) , \beta(\vk) \} = 0 \qtq{and} \{ P , \beta(\vk) \} = 0 \end{equation}

\begin{align} F:= \bigl[ |m|^2 + m + \ol{m}\bigr] \bigl[|n|^2 + n + \ol{n}\bigr]' - \bigl[ |m|^2 + m + \ol{m}\bigr]' \bigl[|n|^2 + n + \ol{n}\bigr]. \end{align}

\begin{align*} F=G+\ol{G}+K+\ol{K} \qtq{where} G &= [ \ol{m}n + \ol{m} + n] \bigl[ (m+1)\ol{n}' - m'(\ol{n}+1) \bigr]\\ \qtq{and} K &= (m -n)(\ol{m}+\ol{n})' . \end{align*}

\begin{equation} \begin{aligned} (m+1)\ol{n}' - m'(\ol{n}+1) &= i \h (m\ol{n}+m+\ol{n})' -(\kappa - \vk)\h (m\ol{n}+m+\ol{n}) \\ &\quad{} + i (\kappa + \vk - 2 q)(m\ol{n}+m+\ol{n}) - 2 i q - i c , \end{aligned} \end{equation}

\begin{align} \int\! (G + \overline G) \, dx = i\int (2q+c) [ m\ol{n} - \ol{m}n + m - \ol{m} - n + \ol{n}] \,dx. \end{align}

\begin{align} \int\! (G + \overline G) \, dx = 2i \int q [ m\ol{n} - \ol{m}n] \,dx. \end{align}

\begin{align*} \int (K + \ol{K}) \, dx &= 2 \int m \ol{n}' + \ol{m} n' \,dx \\ &= 2i \int (\vk - q) [m \ol{n} - \ol{m} n] \,dx - 2 i \int q[m-\ol{m}]\,dx. \end{align*}

\begin{align} \int (K + \ol{K}) \, dx &= - 2i \int q [m \ol{n} - \ol{m} n] \,dx . \end{align}

\begin{align*} &\{\beta(\vk),P(q)\} = \int \big( |n|^2 + n + \ol n \big) q'\, dx = \int - [(1+\ol n)n' + (1+n)\ol n']q\, dx. \end{align*}

\begin{align*} \{\beta(\vk),P(q)\} &= i \int \vk[ n(\ol n+1)-\ol n(n+1) ]q \,dx \\ &- i \int (\ol n+1)q \cdot C_+ (n+1)q - (n+1)q \cdot C_- (\ol n+1)q \,dx \\ & =0.\qedhere \end{align*}

\begin{equation} 2q = \tfrac{1}{w+\kappa} \h (w') + \h \big( \tfrac{w'}{w+\kappa} \big) + \tfrac{2\kappa w}{w+ \kappa} . \end{equation}

\begin{equation} w = \kappa \tfrac{\del\beta}{\del q} = \kappa \big( |m|^2+ m + \ol m \big) \end{equation}

\begin{align*} 2q&= \tfrac{\h(|m|^2+m+\ol m)'}{|m+1|^2} + i\Bigl[ \tfrac{\ol{m}'}{\ol m +1} - \tfrac{m'}{m+1} \Bigr] + 2\kappa \tfrac{|m|^2+m+ \ol m}{|m+1|^2}\\ &=\tfrac{\h(|m|^2+m+\ol m)'}{|m+1|^2} + i\h\Bigl[ \tfrac{\ol{m}'}{\ol m +1} + \tfrac{m'}{m+1} \Bigr]+ 2\kappa \tfrac{|m|^2+m+ \ol m}{|m+1|^2}, \end{align*}

\begin{align} 1 + u(x) = [1+\mu(x)] [1+\ol\mu(x)] \qtq{with} \mu \in H^{s+1}_+(\R). \end{align}

\begin{align*} (1+u) \h \big[ \tfrac{u'}{1+u} \big] = |\mu+1|^2 \h \big( \tfrac{\ol\mu'}{1+\ol\mu} + \tfrac{\mu'}{1+\mu} \big) = i(1+\mu)\ol\mu' - i\mu' (\ol\mu+1) . \end{align*}

\begin{equation} 2q [1+\mu] [1+\ol\mu] = 2C_-\bigl[ i(1+\mu)\ol\mu'\bigr] - 2C_+\bigl[ i\mu' (\ol\mu+1) \bigr] + 2\kappa [\mu+\ol\mu+|\mu|^2] . \end{equation}

\begin{equation} C_+\Bigl[ (1+\ol\mu)\bigl( - i\mu' - C_+(q\mu) + \kappa\mu - q_+ \bigr) \Bigr] =0 . \end{equation}

\begin{equation} - i\mu' - C_+(q\mu) + \kappa\mu - q_+ = \tfrac{f}{1+\ol\mu} \in H^s_-(\R) . \end{equation}

\begin{equation} \int w(x;\kappa, q) \,dx = 2\beta(\kappa) - \kappa\frac{\partial\beta}{\partial\kappa}+ 2\int q \,dx . \end{equation}

\begin{align} \alpha(\kappa;q):= \sum_{\ell\geq 2} \tfrac1\ell\tr\bigl\{ (R_0(\kappa)C_+ q)^\ell \bigr\}, \end{align}

\begin{align*} \|\sqrt{R_0(\kappa)}C_+ q\sqrt{R_0(\kappa)}\|_{\textrm{HS}}^2 &= \frac{1}{2\pi} \int_0^\infty\!\!\int_0^\infty \frac{|\widehat q(\xi-\eta)|^2\,d\eta\,d\xi}{(\eta+\kappa)(\xi+\kappa)}\\ &=\frac{1}{2\pi} \int_\R \frac{\log(1+ \frac{|\xi|}\kappa)}{|\xi|}|\widehat q(\xi)|^2\,d\xi \lesssim \kappa^{-4\eps} \|q\|_{H^s_\kappa}^2<1, \end{align*}

\begin{align*} \alpha(\kappa;q)= \tfrac1{2\pi}\int_0^\infty \tfrac{\beta(\kappa+\xi;q)}{\kappa+\xi}\, d\xi \quad\text{on $\R$} \qquad\text{and} \alpha(\kappa;q)= \sum_{\xi\in 2\pi \Z_+}\tfrac{\beta(\kappa+\xi;q)}{\kappa+\xi} \quad\text{on $\T$}, \end{align*}

\begin{align*} \frac1\ell &\tr\bigl\{ (R_0(\kappa)C_+ q)^\ell \bigr\}\\ &= \sum_{\xi_1, \ldots, \xi_\ell\in 2\pi\Z_+} \frac1\ell \, \frac{\widehat q(\xi_1-\xi_2)}{\kappa+\xi_1} \frac{\widehat q(\xi_2-\xi_3)}{\kappa+\xi_2} \cdots \frac{\widehat q(\xi_\ell-\xi_1)}{\kappa+\xi_\ell}\\ &=\sum_{\substack{\xi_1\leq\min\{\xi_2, \ldots, \xi_\ell\} \\[0.5ex] \xi_1, \ldots, \xi_\ell\in 2\pi\Z_+}} \frac{\widehat q(\xi_1-\xi_2)}{\kappa+\xi_1} \frac{\widehat q(\xi_2-\xi_3)}{\kappa+\xi_2} \cdots \frac{\widehat q(\xi_\ell-\xi_1)}{\kappa+\xi_\ell}\\ &=\sum_{\xi\in 2\pi\Z_+} \frac1{\kappa+\xi} \sum_{\substack{ \eta_2, \ldots, \eta_\ell \in 2\pi\Z \\[0.5ex] \eta_j+ \cdots + \eta_\ell\geq 0, \forall 2\leq j\leq \ell}} \widehat q \bigl(-(\eta_2+\cdots\eta_\ell)\bigr) \prod_{j=2}^\ell\frac{\widehat q(\eta_j)}{\kappa+\xi + \eta_j+ \cdots + \eta_\ell }\\ &=\sum_{\xi\in 2\pi\Z_+} \frac1{\kappa+\xi} \Bigl\langle q,\, \bigl(R_0(\kappa+\xi) C_+ q\bigr)^{\ell-2}R_0(\kappa+\xi) q_+\Bigr\rangle. \end{align*}