Full Counting Statistics of Energy Transfers in Inhomogeneous Nonequilibrium States of $(1+1)D$ CFT

Abstract

Employing the conformal welding technique, we obtain a universal expression for the Full Counting Statistics of energy transfers in a class of inhomogeneous nonequilibrium states of a (1+1)-dimensional unitary Conformal Field Theory. The expression involves the Schwarzian action of a complex field obtained by solving a Riemann–Hilbert type problem related to conformal welding of infinite cylinders. On the way, we establish a formula for the extension of characters of unitary positive-energy representations of the Virasoro algebra to 1-parameter groups of circle diffeomorphisms and we develop techniques, based on the analysis of certain classes of Fredholm operators, that allow to control the leading asymptotics of such an extension for small modular parameters $\,\tau$.

Appendices

Appendix A

In a conformal field theory on a circle of circumference $\,L$, the energy-momentum components with Euclidian time dependence are

$$\begin{aligned} T_\pm (x\pm \mathrm {i}vt)=\frac{_{2\pi }}{^{L^2}} \sum \limits _{n=-\infty }^\infty \mathrm{e}^{\pm \frac{2\pi \mathrm {i}n}{L}(x\pm \mathrm {i}vt)}\big (L^\pm _n-\frac{_c}{^{24}}\delta _{n,0}\big ), \end{aligned}$$

(A.1)

where $\,L^\pm _n\,$ are generators of two commuting unitary Virasoro representation in the space of states. They satisfy the relations

$$\begin{aligned} T_\pm (x\pm \mathrm {i}vt)\,=\,\mathrm{e}^{tH_L}T_\pm (x)\,\mathrm{e}^{-tH_L}, \end{aligned}$$

(A.2)

where the Hamiltonian of the theory

$$\begin{aligned} H_L\,=\,v\int _0^L\big (T(x)+\bar{T}(-x)\big )\,dx \,=\,\frac{_{2\pi v}}{^L}\big (L_0+\bar{L}_0-\frac{_c}{^{12}}\big ). \end{aligned}$$

(A.3)

The normalized vacuum vector $\,\big |0\big \rangle \,$ is the state for which $\,L_n\big |0\big \rangle =0=\bar{L}_n\big |0\big \rangle \,$ for $\,n\ge 0$. It follows then by a straightforward calculation that

$$\begin{aligned} \big \langle 0\big |\,T_\pm (x\pm \mathrm {i}vt)\big |0\big \rangle \, =\,-\frac{_{\pi c}}{^{12L^2}} \end{aligned}$$

(A.4)

with the vacuum energy $\,\langle 0|H_L|0\rangle =-\frac{\pi c v}{6L}\,$ and that for $\,t_1\not =t_2$,

$$\begin{aligned} \big \langle 0\big |\,\mathcal{T}\big (T_\pm (x_1\pm \mathrm {i}vt_1) \,T_\pm (x_2\pm \mathrm {i}vt_2)\big ) \big |0\big \rangle= & {} \big (\frac{_{\pi c}}{^{12L^2}}\big )^{2} +\frac{_{2\pi ^2c}}{^{L^4}}\frac{_{z_1^2z_2^2}}{^{(z_1-z_2)^4}} \end{aligned}$$

(A.5)

where $\,z_1=\mathrm{e}^{\frac{2\pi \mathrm {i}}{L}(x_1\pm \mathrm {i}vt_1)}\,$ and $\,z_2 =\mathrm{e}^{\frac{2\pi \mathrm {i}}{L}(x_2\pm \mathrm {i}vt_2)}$.

Appendix B

We prove here that the finite-volume transient fluctuation relation (6.15) implies the infinite-volume one (8.65). First, let us note that in view of (6.15) and the uniform bound $|\Psi _{t,L}(\lambda )|\le 1$ on the real axis, the holomorphic extension of $\,\Psi _{t,L}\,$ from the real axis to the strip $\,0<\frac{\mathrm{Im}(\lambda )}{\Delta \beta }<1\,$ satisfies for $\,b>|\Delta \beta |\,$ the Cauchy-type formula

$$\begin{aligned} \Psi _{t,L}(\lambda )=\frac{\mathrm{sgn}(\Delta \beta )}{2\pi \mathrm {i}b} \int _{-\infty }^{\infty } \bigg (\frac{\Psi _{t,L}(\mu )}{\sinh \big (\frac{\mu -\lambda }{b}\big )}- \frac{\Psi _{t,L}(-\mu )}{\sinh \big (\frac{\mu +\mathrm {i}\Delta \beta -\lambda }{b}\big )} \bigg )d\mu \,. \end{aligned}$$

(B.1)

Besides the boundary values $\,\Psi _{t,L}(\lambda )\,$ and $\,\Psi _{t,L}(\lambda +\mathrm {i}\Delta \beta )=\Psi _{t,L}(-\lambda )\,$ for $\,\lambda \,$ real satisfy the equations

$$\begin{aligned} \frac{_1}{^2}\Psi _{t,L}(\lambda )= & {} \frac{\mathrm{sgn}(\Delta \beta )}{2\pi \mathrm {i}b} \,PV\int _{-\infty }^{\infty }\frac{\Psi _{t,L}(\mu )}{\sinh \big (\frac{\mu -\lambda }{b}\big )}d\mu \,\nonumber \\&-\,\frac{\mathrm{sgn}(\Delta \beta )}{2\pi \mathrm {i}b}\int _{-\infty }^{\infty } \frac{\Psi _{t,L}(-\mu )}{\sinh \big (\frac{\mu +\mathrm {i}\Delta \beta -\lambda }{b}\big )}d\mu \end{aligned}$$

(B.2)

and

$$\begin{aligned} \frac{_1}{^2}\Psi _{t,L}(-\lambda )= & {} \frac{\mathrm{sgn}(\Delta \beta )}{2\pi \mathrm {i}b} \int _{-\infty }^{\infty }\frac{\Psi _{t,L}(\mu )}{\sinh \big (\frac{\mu -\lambda -\mathrm {i}\Delta \beta }{b}\big )}d\mu \,\nonumber \\&-\,\frac{\mathrm{sgn}(\Delta \beta )}{2\pi \mathrm {i}b}\,PV\int _{-\infty }^{\infty } \frac{\Psi _{t,L}(-\mu )}{\sinh \big (\frac{\mu -\lambda }{b}\big )}d\mu , \end{aligned}$$

(B.3)

where $\,PV\,$ stands for principal value. Taking the limit $\,L\rightarrow \infty \,$ on the right-hand side of (B.1) using the pointwise convergence of $\,\Psi _{t,L}\,$ to $\,\Psi _t\,$ on the real axis, we obtain a function holomorphic on the open strip $\,0<\frac{\mathrm{Im}(\lambda )}{\Delta \beta }<1\,$ that we shall also denote $\,\Psi _t(\lambda )$.  Using again the pointwise convergence of $\,\Psi _{t,L}$ to $\,\Psi _{t}\,$ and the uniform convergence of $\,\Psi '_{t,L}\,$ to $\,\Psi '_t\,$ on bounded subsets of the real axis, see Remark 2 after (8.63), we obtain from (B.2) and (B.3) the identities that have the same form but with the real axis $\,\Psi _{t,L}\,$ replaced by $\,\Psi _t$. They guarantee that the function $\,\Psi _t\,$ holomorphic in the open strip obtained from the limit of (B.1) has $\,\Psi _t(\lambda )\,$ and $\,\Psi _t(-\lambda )\,$ as the boundary values at real $\,\lambda \,$ and at $\,\lambda +\mathrm {i}\Delta \beta$,  respectively.  This establishes the transient fluctuation relation (8.65).

Appendix C

From the residue theorem,

$$\begin{aligned}&\int \frac{\mathrm{e}^{-\mathrm {i}py}}{\sinh ^4\big (\frac{\pi }{\gamma }(y\pm \mathrm {i}0)\big )}dy -\int \frac{\mathrm{e}^{-\mathrm {i}p(y\mp \gamma \mathrm {i})}}{\sinh ^4\big (\frac{\pi }{\gamma }(y\pm \mathrm {i}0) \big )}dy=\mp \frac{_{\pi \mathrm {i}}}{^3}\,\partial _z^3\Big |_{z=0}\, \frac{z^4\mathrm{e}^{-ipz}}{\sinh ^4\big (\frac{\pi }{\gamma }z\big )}\nonumber \\&\quad =\mp \frac{_{\pi \mathrm {i}}}{^3}\frac{_{\gamma ^4}}{^{\pi ^4}}\, \partial _z^3\Big |_{z=0} \frac{e^{-\mathrm {i}pz}}{(1+\frac{1}{6}\frac{\pi ^2}{\gamma ^2}z^2)^4} =\mp \frac{_{\pi \mathrm {i}}}{^3}\frac{_{\gamma ^4}}{^{\pi ^4}}\, \partial _z^3\Big |_{z=0}\Big ( e^{-\mathrm {i}pz}\big (1-\frac{_2}{^3}\frac{_{\pi ^2}}{^{\gamma ^2}}z^2\big )\Big )\nonumber \\&\quad =\mp \frac{_{\pi \mathrm {i}}}{^3}\frac{_{\gamma ^4}}{^{\pi ^4}}\, \partial _z^2\Big |_{z=0}\Big ( (-\mathrm {i}p)\,e^{-\mathrm {i}pz}\big (1-\frac{_2}{^3}\frac{_{\pi ^2}}{^{\gamma ^2}}z^2\big ) -\frac{_4}{^3}\mathrm{e}^{-\mathrm {i}pz}\frac{_{\pi ^2}}{^{\gamma ^2}}z\Big )\nonumber \\&\quad =\mp \frac{_{\pi \mathrm {i}}}{^3}\frac{_{\gamma ^4}}{^{\pi ^4}}\, \partial _z\Big |_{z=0}\Big ( (-\mathrm {i}p)^2e^{-\mathrm {i}pz} -\frac{_{8}}{^3}\frac{_{\pi ^2}}{^{\gamma ^2}}(-\mathrm {i}p)\,e^{-\mathrm {i}pz}z -\frac{_4}{^3}\mathrm{e}^{-\mathrm {i}pz}\frac{_{\pi ^2}}{^{\gamma ^2}}\Big )\nonumber \\&\quad =\mp \frac{_{\pi \mathrm {i}}}{^3}\frac{_{\gamma ^4}}{^{\pi ^4}}\, \Big ((-\mathrm {i}p)^3-\frac{_{4\pi ^2}}{^{\gamma ^2}}(-\mathrm {i}p)\Big )=\pm \frac{_{1}}{^3}\frac{_{\gamma ^4}}{^{\pi ^3}}\,p \big (p^2+\frac{_{4\pi ^2}}{^{\gamma ^2}}\big ). \end{aligned}$$

(C.1)

Hence

$$\begin{aligned} \int \frac{\mathrm{e}^{-\mathrm {i}py}}{\sinh ^4\big (\frac{\pi }{\gamma }(y\pm \mathrm {i}0)\big )}dy= \pm \frac{_{\gamma ^4}}{^{3\pi ^3}}\frac{_{p\big (p^2+\frac{_{4\pi ^2}}{^{\gamma ^2}}\big )}}{1-\mathrm{e}^{\mp \gamma p}}. \end{aligned}$$

(C.2)

Appendix D

We collect here few results concerning operators of fast-decay and Schwartz type introduced in Sect. 10.2 in Definitions 1 to 4. The two cases will be covered separately as they often differ and both are needed in the main text.

1.1 D.1 Products of operators of fast-decay and Schwartz type

Let us start by two Propositions that are straightforward to prove.

Proposition D1

If $\,\mathcal{D}_1,\mathcal{D}_2\,$ are operators on $\,L^2({\mathbb R})\,$ of fast-decay type then so is their product $\,\mathcal{D}_1\mathcal{D}_2$. If $\,\mathcal{D}_{1,L},\mathcal{D}_{2,L}\,$ are families of fast-decay type operators converging with speed $\,L^{-1}\,$ to fast-decay type operators $\,\mathcal{D}_1,\mathcal{D}_2$,  respectively,  then the products $\,\mathcal{D}_{1,L}\mathcal{D}_{2,L}\,$ converge with speed $\,L^{-1}\,$ to the product $\,\mathcal{D}_1\mathcal{D}_2\,$ as operators of fast-decay type. If $\,D_1,D_2\,$ are operators on $\,L^2_0(S^1_L)\,$ of fast-decay type then so is their product $\,D_{1}D_{2}$.

Proposition D2

If $\,\mathcal{D}_1,\mathcal{D}_2\,$ are operators on $\,L^2({\mathbb R})\,$ of Schwartz $\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}'$ and $\widehat{\mathcal{J}}'\times \widehat{\mathcal{J}}''$ type, respectively, then $\,\mathcal{D}_1\mathcal{D}_2\,$ is of Schwartz $\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}''$ type. If $\,\mathcal{D}_{1,L},\mathcal{D}_{2,L}\,$ are families of such operators converging with speed $\,L^{-1}\,$ to $\,\mathcal{D}_1,\mathcal{D}_2$,  respectively,  then $\,\mathcal{D}_{1,L}\mathcal{D}_{2,L}\,$ converges with speed $\,L^{-1}\,$ to $\,\mathcal{D}_1\mathcal{D}_2\,$ as operators of Schwartz $\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}''$ type. If $\,D_1,D_2\,$ are operators on $\,L^2_0(S^1_L)\,$ of Schwartz $\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}'$ and $\widehat{\mathcal{J}}'\times \widehat{\mathcal{J}}''$ type, respectively, then $\,D_1D_2\,$ is of Schwartz $\widehat{\mathcal{J}}\times \widehat{\mathcal{J}}''$ type.

The next result is a little more subtle.

Proposition D3

If $\,D_{1,L},D_{2,L}\,$ are families of operators on $\,L^2(S^1_L)\,$ of Schwartz ${\mathbb R}_{\sigma }\times {\mathbb R}_{\sigma '}$ and ${\mathbb R}_{\sigma '}\times {\mathbb R}_{\sigma ''}$ type for $\,\sigma ,\sigma ',\sigma '' =\pm \,$ converging with speed $\,L^{-1}\,$ to operators $\,\mathcal{D}_1,\mathcal{D}_2\,$ on $\,L^2_0({\mathbb R})\,$ of the same Schwartz type then the operators $\,D_{1,L}D_{2,L}\,$ on $\,L^2_0(S^1_L)\,$ of Schwartz ${\mathbb R}_{\sigma }\times {\mathbb R}_{\sigma ''}$ type converge with speed $\,L^{-1}\,$ to $\,\mathcal{D}_{1}\mathcal{D}_{2}$.

Proof

Let $\,\mathcal{D}_{1,L}\,$ and $\,\mathcal{D}_{2,L}\,$ be operators on $\,L^2({\mathbb R})\,$ of Schwartz ${\mathbb R}_{\sigma }\times {\mathbb R}_{\sigma '}$ and and ${\mathbb R}_{\sigma '}\times {\mathbb R}_{\sigma ''}$ type, respectively, converging with speed $\,L^{-1}\,$ to, respectively, $\,\mathcal{D}_1\,$ and $\,\mathcal{D}_2\,$ and such that $\,D_{1,L}\,$ and $\,D_{2,L}\,$ are the L-periodization of, respectively, $\,\mathcal{D}_{1,L}\,$ and $\,\mathcal{D}_{2,L}\,$ (the existence of such operators follows from our assumptions in view of Definition 4 of Sect. 10.2). Let $\,\mathcal{D}_{3,L}\,$ be the operators on $\,L^2({\mathbb R})\,$ with the momentum-space kernels

$$\begin{aligned} \widehat{\mathcal{D}_{3,L}}(p,q)=\frac{1}{L}\sum \limits _{p_n\in {\mathbb R}_{\sigma '}} \widehat{\mathcal{D}_{1,L}}(p,p_n)\,\widehat{\mathcal{D}_{2,L}}(p_n,q)\,. \end{aligned}$$

(D.1)

Note that the product operators $\,D_{3,L}=D_{1,L}D_{2,L}\,$ are the L-periodization of $\,\mathcal{D}_{3,L}$.  Let $\,\mathcal{D}_3=\mathcal{D}_1\mathcal{D}_2\,$ with the momentum-space kernel

$$\begin{aligned} \widehat{\mathcal{D}_3}(p,q)=\frac{1}{2\pi }\int _{{\mathbb R}_{\sigma '}}\widehat{\mathcal{D}_1}(p,r)\, \widehat{\mathcal{D}_2}(r,q)\,dr\,. \end{aligned}$$

(D.2)

We shall prove Proposition D3 by showing that $\,\mathcal{D}_{3,L}\,$ converge with speed $\,L^{-1}\,$ to $\,\mathcal{D}_3\,$ as operators of Schwartz ${\mathbb R}_{\sigma }\times {\mathbb R}_{\sigma ''}$ type. To this end, let us estimate for $\,(p,q)\in {\mathbb R}_{\sigma }\times {\mathbb R}_{\sigma ''}$

$$\begin{aligned}&\Big |\partial _p^{\ell _1}\partial _q^{\ell _2}\widehat{\mathcal{D}_{3,L}}(p,q)- \partial _p^{\ell _1}\partial _q^{\ell _2}\widehat{\mathcal{D}_3}(p,q)\Big |\nonumber \\&\quad \le \,\frac{1}{L}\sum \limits _{p_n\in {\mathbb R}_{\sigma '}}\Big |\partial _p^{\ell _1} \widehat{\mathcal{D}_{1,L}}(p,p_n)\,\partial _q^{\ell _2}\widehat{\mathcal{D}_{2,L}}(p_n,q) -\partial _p^{\ell _1}\widehat{\mathcal{D}_1}(p,p_n)\,\partial _q^{\ell _2}\widehat{\mathcal{D}_2}(p_n,q)\Big |\nonumber \\&\qquad +\Big |\frac{1}{L}\sum \limits _{p_n\in {\mathbb R}_{\sigma '}} \partial _p^{\ell _1}\widehat{\mathcal{D}_1}(p,p_n)\, \partial _q^{\ell _2}\widehat{\mathcal{D}_2}(p_n,q)- \frac{1}{2\pi }\int _{{\mathbb R}_{\sigma '}}\partial _p^{\ell _1}\widehat{\mathcal{D}_1}(p,r)\, \partial _q^{\ell _2}\widehat{\mathcal{D}_2}(r,q)\,dr\Big |. \nonumber \\ \end{aligned}$$

(D.3)

The $1\mathrm{st}$ term on the right is easily bounded using the convergence of $\,\mathcal{D}_{i,L}\,$ to $\,\mathcal{D}_i\,$ by

$$\begin{aligned} \frac{L^{-1}C_{\ell _1,\ell _2,k}}{(1+p^2)^k(1+q^2)^k}\, \sum \limits _{p_n\in {\mathbb R}_{\sigma '}}\frac{1}{L}\,\frac{1}{(1+p_n^2)^{2k}} \ \le \ \frac{L^{-1}C'_{\ell _1,\ell _2,k}}{(1+p^2)^k(1+q^2)^k}\,. \end{aligned}$$

(D.4)

The $2\mathrm{nd}$ term on the right-hand side of (D.3) is estimated by

$$\begin{aligned}&\frac{1}{2\pi }\sum \limits _{\widehat{J}_n\subset {\mathbb R}_{\sigma '}}\int _{\widehat{J}_n} \Big |\partial _p^{\ell _1}\widehat{\mathcal{D}_1}(p,p_n)\, \partial _q^{\ell _2}\widehat{\mathcal{D}_2}(p_n,q)-\partial _p^{\ell _1}\widehat{\mathcal{D}_1}(p,r)\, \partial _q^{\ell _2}\widehat{\mathcal{D}_2}(r,q)\Big |\,dr\nonumber \\&\quad +\,\frac{1}{2\pi }\int _{\widehat{J}_0\cap {\mathbb R}_{\sigma '}}\Big | \partial _p^{\ell _1}\widehat{\mathcal{D}_1}(p,r)\, \partial _q^{\ell _2}\widehat{\mathcal{D}_2}(r,q)\Big |\,dr \end{aligned}$$

(D.5)

for

$$\begin{aligned} \widehat{J}_n=\Big ]\frac{_{2\pi (n-\frac{1}{2})}}{^L}, \frac{_{2\pi (n+\frac{1}{2})}}{^L}\Big ] \end{aligned}$$

(D.6)

so that $\,p_n\,$ is the middle-point of $\,\widehat{J}_n$.  The $1\mathrm{st}$ line is estimated by (D.4) using the bounds of the $\,r$-derivative of $\,\partial _p^{\ell _1}\widehat{\mathcal{D}_1}(p,r)\, \partial _q^{\ell _2}\widehat{\mathcal{D}_2}(r,q)\,$ and the $2\mathrm{nd}$ line using the bounds on that function and the small length $\,|\widehat{J}_0\cap {\mathbb R}_{\sigma '}|=\pi L^{-1}$.  Altogether, the left-hand side of (D.3) is then bounded by $\,L^{-1}C_{\ell _1,\ell _2,k}(1+p^2)^{-k}(1+q^2)^{-k}\,$ for some $\,L$-independent constants $\,C_{\ell _1,\ell _2,k}$,  as required. $\quad \square$

1.2 D.2 Fredholm determinants

Let $\,\mathcal{D}\,$ be the operator of fast-decay type on $\,L^2({\mathbb R})$. Then $\,\mathcal{I}+\mathcal{D}\,$ is a Fredholm operator and its determinant may be defined by the series [27]

$$\begin{aligned} \det (\mathcal{I}+\mathcal{D})=\sum \limits _{r=0}^\infty \frac{1}{{r!(2\pi )^r}}\int _{{\mathbb R}^r} {\det }_{r\times r}\big (\widehat{\mathcal{D}}(q_i,q_j)\big )\,dq_1\cdots dq_r\,. \end{aligned}$$

(D.7)

The determinant of an $\,r\times r\,$ matrix $\,M=(M_{ij})\,$ may be viewed as an r-linear function $\,d_r(m_1,\dots ,m_r)\,$ of the row vectors of $\,M$,  where $\,(m_i)_j=M_{ij}$.  We shall frequently use below the Hadamard inequality that states that

$$\begin{aligned} \big |{\det }_{r\times r}(M)\big |\,\le \,\prod _{i=1}^r\Vert m_i\Vert \,, \end{aligned}$$

(D.8)

where $\,\Vert m\Vert \,$ stands for the Euclidian norm of the vector $\,m$.  In particular, we infer that

$$\begin{aligned} \big |{\det }_{r\times r}\big (\widehat{\mathcal{D}}(q_i,q_j)\big )\big |\,\le \, \prod \limits _{i=1}^r\Big (\sqrt{r}\,\frac{C_k}{(1+q_i^2)^{k}}\Big ) \,=\, r^{\frac{r}{2}}C_k^r\prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^k} \end{aligned}$$

(D.9)

which assures the convergence of the series (D.7).

Proposition D4

Let $\,\mathcal{D}_L\,$ and $\,\mathcal{D}\,$ be operators on $\,L^2({\mathbb R})\,$ of fast-decay type such that $\,\mathcal{D}_L\,$ converge to $\,\mathcal{D}\,$ with speed $\,L^{-1}$. Then

$$\begin{aligned} \big |\det (\mathcal{I}+\mathcal{D}_L)-\det (\mathcal{I}+\mathcal{D})\big |\,\le \,\,L^{-1}C \end{aligned}$$

(D.10)

for some $\,L$-independent constant $\,C$.

Proof

Viewing the determinant as the r-linear function of row vectors, we may write

$$\begin{aligned}&{\det }_{r\times r}\big (\widehat{\mathcal{D}_L}(q_i,q_j)\big )-{\det }_{r\times r} \big (\widehat{\mathcal{D}}(q_i,q_j)\big )\nonumber \\&\quad =\sum \limits _{k=1}^r d_r(m_{1,L},\dots ,m_{k-1,L}, m_{k,L}-m_k,m_{k+1},\dots ,m_r)\,, \end{aligned}$$

(D.11)

where $\,(m_{i,L})_j=\widehat{\mathcal{D}_L}(q_i,q_j)\,$ and $\,(m_{i})_j =\widehat{\mathcal{D}}(q_i,q_j)$.  Then, by the Hadamard inequality,

$$\begin{aligned}&\big |{\det }_{r\times r}\big (\widehat{\mathcal{D}_L}(q_i,q_j)\big )-{\det }_{r\times r} \big (\widehat{\mathcal{D}}(q_i,q_j)\big )\big |\nonumber \\&\quad \le \sum \limits _{k=1}^r \Big (\prod \limits _{i=1}^{k-1}\Vert m_{i,L}\Vert \Big )\Vert m_{k,L}-m_k\Vert \Big (\prod \limits _{i=k+1}^r\Vert m_i\Vert \Big )\nonumber \\&\quad \le r\,L^{-1}\prod \limits _{i=1}^r\frac{\sqrt{r}\, C_k}{(1+q_i^2)^k}\,=\, L^{-1}r^{\frac{r}{2}+1}C_k^r\prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^k}\,. \end{aligned}$$

(D.12)

The assertion of Proposition D4 follows now from the Fredholm series representation (D.7) for $\,\det (\mathcal{I}+\mathcal{D}_L)\,$ and $\,\det (\mathcal{I}+\mathcal{D})$. $\quad \square$

If $\,D\,$ is an operator on $\,L^2_0(S^1_L)\,$ of fast-decay type in the sense of Definition 4 of Sect. 10.2 then $\,I+D\,$ is a Fredholm operator and its determinant may be defined by the series

$$\begin{aligned} \det (I+D)= & {} \sum \limits _{r=0}^\infty \frac{1}{r!}\sum _{(n_1,\dots n_r)\in {\mathbb Z}_{\not =0}^{\ r}} {\det }_{r\times r}\big (D_{n_i,n_j}\big )\nonumber \\= & {} \sum \limits _{r=0}^\infty \frac{_1}{r!\,L^r} \sum _{(n_1,\dots n_r)\in {\mathbb Z}_{\not =0}^{\ r}} {\det }_{r\times r}\big (\widehat{\mathcal{D}}(p_{n_i},p_{n_j})\big ) \end{aligned}$$

(D.13)

if $\,\mathcal{D}\,$ is a fast-decay operator on $\,L^2({\mathbb R})\,$ such that $\,D\,$ is its L-periodization. The convergence of the series follows from the Hadamard inequality that implies the bound

$$\begin{aligned} \big |{\det }_{r\times r}\big (\widehat{\mathcal{D}}(p_{n_i},p_{n_j})\big |\,\le \, r^{\frac{r}{2}}\,C_k^r\prod \limits _{i=1}^r\frac{1}{(1+p_{n_i}^2)^k} \end{aligned}$$

(D.14)

and the uniform in L convergence of the series $\,\sum \limits _{0\not =n\in {\mathbb Z}} \frac{1}{L}\,\frac{1}{(1+p_n^2)^k}\,$ for $\,k\ge 1$.

Proposition D5

Let $\,D_L\,$ be operators on $\,L^2(S^1_L)\,$ of fast-decay type and let $\,\delta D_L\,$ be similar operators converging with speed $\,L^{-1}\,$ to zero. Suppose that $\,D_L\,$ are the L-periodization of operators $\,\mathcal{D}_L\,$ on $\,L^2({\mathbb R})\,$ of fast-decay type satisfying uniform in $\,L\,$ fast-decay bounds.  Then for ${\widetilde{D}}_L=D_L+\delta D_L$,

$$\begin{aligned} \big |\det (I+\widetilde{D}_L)-\det (I+D_L)\big |\,\le \,L^{-1}C \end{aligned}$$

(D.15)

for some $\,L$-independent constant $\,C$.

Proof

Let $\,\delta \mathcal{D}_L\,$ be fast-decay operators on $\,L^2({\mathbb R})\,$ converging with speed $\,L^{-1}\,$ to zero and such that $\,\delta D_L\,$ are their L-periodization (their existence follows from Definition 4 of Sect. 10.3). Set $\,\widetilde{\mathcal{D}}_L=\mathcal{D}_L+\delta \mathcal{D}_L$. Then

$$\begin{aligned}&\Big |\det (I+\widetilde{D}_L)-\det (I+D_L)\Big |\nonumber \\&\quad \le \,\sum \limits _{r=0}^\infty \frac{1}{r!\,L^r}\sum \limits _{(n_1,\dots ,n_r)\in {\mathbb Z}_{\not =0}^{\ r}} \Big |{\det }_{r\times r}\big (\widehat{\widetilde{\mathcal{D}}_L}(p_{n_i},p_{n_j}))\big )- {\det }_{r\times r}\big (\widehat{\mathcal{D}_L}(p_{n_i},p_{n_j})\Big |. \nonumber \\ \end{aligned}$$

(D.16)

Using the Hadamard inequality as in Proof of Proposition D4 above, we obtain the bound

$$\begin{aligned} \big |{\det }_{r\times r}\big (\widehat{\widetilde{\mathcal{D}}_L}(p_{n_i},p_{n_j})\big )- {\det }_{r\times r}\big (\widehat{\mathcal{D}_L}(p_{n_i},p_{n_j})\big |\,\le \, L^{-1}r^{\frac{r}{2}+1}\,C_k^r\prod \limits _{i=1}^r\frac{1}{(1+p_{n_i}^2)^k}\nonumber \\ \end{aligned}$$

(D.17)

from which (D.15) follows. $\quad \square$

Corollary D1

Let $\,D_L\,$ be operators on $\,L^2_0(S^1_L)\,$ of fast-decay type converging with speed $\,L^{-1}\,$ to operator $\,\mathcal{D}\,$ on $\,L^2({\mathbb R})\,$ of fast-decay type and let $\,D\,$ be the L-periodization of $\,\mathcal{D}$.  Then

$$\begin{aligned} \big |\det (I+D_L)-\det (I+D)\big |\,\le \,L^{-1}C \qquad \end{aligned}$$

(D.18)

for some $\,L$-independent constant $\,C$.

Proof

We set $\,D'_L=D\,$ and $\,\delta D'_L=D_L-D\,$ and apply Proposition D5 to the pair $\,(D'_L,\delta D'_L)$. $\quad \square$

Proposition D6

If $\,\mathcal{D}\,$ is an operator on $\,L^2({\mathbb R})\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type and $\,D\,$ is its L-periodization then

$$\begin{aligned} \big |\det (I+D)-\det (\mathcal{I}+\mathcal{D})\big |\,\le \,L^{-1}C \end{aligned}$$

(D.19)

for some $\,L$-independent constant $\,C$.

Proof

Let $\,\mathcal{D}_L\,$ be the operators on $\,L^2({\mathbb R})\,$ with momentum space kernels

$$\begin{aligned} \widehat{\mathcal{D}_L}(p,q)\,=\,\sum \limits _{0\not =m,n\in {\mathbb Z}} {\varvec{1}}_{\widehat{J}_m}(p)\,\widehat{\mathcal{D}}(p_m,p_n)\,{\varvec{1}}_{\widehat{J}_n}(q)\,, \end{aligned}$$

(D.20)

where $\,{\varvec{1}}_{\widehat{J}_n}\,$ is the characteristic function of the interval $\,\widehat{J}_n$, see (D.6).  We have the identity

$$\begin{aligned} \det (I+D)=\det (\mathcal{I}+\mathcal{D}_L) \end{aligned}$$

(D.21)

and for $\,p\in \widehat{J}_m\,$ and $\,q\in \widehat{J}_n\,$ with $\,m,n\not =0$,

$$\begin{aligned} \big |\widehat{\mathcal{D}_L}(p,q)-\widehat{\mathcal{D}}(p,q)\big |\,=\, \big |\widehat{\mathcal{D}}(p_m,p_n)-\widehat{\mathcal{D}}(p,q)\big |\,\le \, \frac{L^{-1}C_k}{(1+p^2)^k(1+q^2)^k} \nonumber \\ \end{aligned}$$

(D.22)

for some $\,C_k\,$ by the Schwartz-type property of $\,\mathcal{D}$.  This bound may fail, however, for $\,p\,$ or $\,q\,$ in $\,\widehat{J}_0\,$ in which case $\,\mathcal{D}_L(p,q)=0\,$ and $\,\mathcal{D}(p,q)\,$ may be of order 1 with a possible discontinuity at $\,p=0\,$ and/or $\,q=0$.  If we define $\,\mathcal{D}'_L\,$ as the operator on $\,L^2({\mathbb R})\,$ with the momentum-space kernel

$$\begin{aligned} \widehat{\mathcal{D}'_L}(p,q)={\varvec{1}}_{{\mathbb R}\setminus \widehat{J}_0}(p)\,\widehat{\mathcal{D}}(p,q)\, {\varvec{1}}_{{\mathbb R}\setminus \widehat{J}_0}(q) \end{aligned}$$

(D.23)

then repeating the argument from Proof of Proposition D4, one shows using the bound (D.22) that

$$\begin{aligned} \big |\det (I+D)-\det (\mathcal{I}+\mathcal{D}'_L)|\,=\, \big |\det (\mathcal{I}+\mathcal{D}_L)-\det (\mathcal{I}+\mathcal{D}'_L)|\,\le \,\frac{_1}{^2}L^{-1}C\,. \nonumber \\ \end{aligned}$$

(D.24)

On the other hand,

$$\begin{aligned}&|\det (\mathcal{I}+\mathcal{D}'_L)-\det (\mathcal{I}+\mathcal{D})\big |\nonumber \\&\quad \le \, \sum \limits _{r=1}^\infty \frac{1}{{(r-1)! (2\pi )^r}}\int _{\widehat{J}_0}dq_1\int _{{\mathbb R}^{r-1}} \big |{\det }_{r\times r}\big (\widehat{\mathcal{D}}(q_i,q_j)\big )\big |\,dq_2\cdots dq_r\nonumber \\&\quad \le \,\sum \limits _{r=1}^\infty \frac{r^{\frac{r}{2}}C_k^r}{{(r-1)!(2\pi )^r}} \int _{\widehat{J}_0}dq_1\int _{{\mathbb R}^{r-1}}\prod \limits _{i=1}^r\frac{1}{(1+q_i^2)}\, dq_2\cdots dq_r\,\le \,\frac{_1}{^2}L^{-1}C\,,\nonumber \\ \end{aligned}$$

(D.25)

where the $\,L^{-1}\,$ factor is due to the length $\,\frac{2\pi }{L}\,$ of $\,\widehat{J}_0$. Together with (D.24) this gives (D.19). $\quad \square$

Corollary D2

If $\,D_L\,$ are operators on $\,L^2_0(S^1_L)\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type converging with speed $\,L^{-1}\,$ to operator $\,\mathcal{D}\,$ on $\,L^2({\mathbb R})\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type then

$$\begin{aligned} \big |\det (I+D_L)-\det (\mathcal{I}+\mathcal{D})\big |\,\le \,L^{-1}C \end{aligned}$$

(D.26)

for some $\,L$-independent constant $\,C$.

Proof

This follows directly from Corollary D1 and Proposition D6 since the Schwartz type convergence implies fast-decay type one. $\quad \square$

1.3 D.3 Inverses of Fredholm operators

Proposition D7

If $\,\mathcal{D}\,$ is a fast-decay type operator on $\,L^2({\mathbb R})\,$ and $\,\det (\mathcal{I}+\mathcal{D})\not =0\,$ then the Fredholm operator $\,\mathcal{I}+\mathcal{D}\,$ is invertible and $\,\mathcal{R}=\mathcal{I}-(\mathcal{I}+\mathcal{D})^{-1}\,$ is of fast-decay type.  If, moreover, operators $\,\mathcal{D}_L\,$ on $\,L^2({\mathbb R})\,$ of fast-decay type converge to $\,\mathcal{D}\,$ with speed $\,L^{-1}\,$ then $\,\mathcal{R}_L=\mathcal{I}-(\mathcal{I}+\mathcal{D}_L)^{-1}\,$ are well defined for $\,L\,$ large enough and are of fast-decay type and they converge to $\,\mathcal{R}\,$ with speed $\,L^{-1}$.

Proof

The invertibility of $\,\mathcal{I}+\mathcal{D}\,$ follows since this operator has no zero eigenvalue and $\,(\mathcal{I}+\mathcal{D})^{-1}\,$ is also a Fredholm operator. The momentum-space kernel of $\,\mathcal{R}\,$ is given by the Fredholm series [27]

$$\begin{aligned} \widehat{\mathcal{R}}(p,q)\,=\,\frac{1}{{\det (\mathcal{I}+\mathcal{D})}} \sum \limits _{r=0}^\infty \frac{1}{{r!(2\pi )^r}} \int _{{\mathbb R}^r}{\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}(q_i,q'_j)\big ) \,\,dq_1\cdots dq_r\,, \nonumber \\ \end{aligned}$$

(D.27)

where

$$\begin{aligned} q_0=p\,,\qquad q'_0=q\,,\qquad q_i=q_i'\quad \ \text {for}\quad \ i=1,\dots ,r\,. \end{aligned}$$

(D.28)

By the Hadamard inequality,

$$\begin{aligned} \Big |{\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}(q_i,q'_j)\big )\Big |\,\le \, (r+1)^{\frac{r+1}{2}}\,C_k^{r+1}\,\frac{1}{(1+p^2)^k} \prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^k} \end{aligned}$$

(D.29)

and similarly, applying it to the column vectors,

$$\begin{aligned} \Big |{\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}(q_i,q'_j)\big )\Big |\,\le \, (r+1)^{\frac{r+1}{2}}\,C_k^{r+1}\,\frac{1}{(1+q^2)^k} \prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^k}\,. \end{aligned}$$

(D.30)

Using the geometric mean of those estimates, we infer that

$$\begin{aligned} \big |\widehat{\mathcal{R}}(p,q)\big |\le & {} \frac{1}{{|\det (\mathcal{I}+\mathcal{D})|}}\nonumber \\&\sum \limits _{r=0}^\infty \frac{{(r+1)^{\frac{r+1}{2}}\,C_{2k}^{r+1}}}{{r!(2\pi )^r}} \frac{1}{(1+p^2)^k(1+q^2)^k} \int _{{\mathbb R}^r}\prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^{2k}}\,dq_1\cdots dq_r\nonumber \\\le & {} \frac{C_k}{(1+p^2)^k(1+q^2)^k} \end{aligned}$$

(D.31)

for some new constants $\,C_k$.  This proves that $\,\mathcal{R}\,$ is of fast-decay type.

Now, if $\,\mathcal{D}_L\,$ converge with speed $\,L^{-1}\,$ to $\,\mathcal{D}\,$ then, by Proposition D4, $\,\det (\mathcal{I}+\mathcal{D}_L)\,$ converges with speed $\,L^{-1}\,$ to $\,\det (\mathcal{I}+\mathcal{D})\,$ and hence is bounded away from zero for $\,L\,$ sufficiently large. On the other hand, using the Hadamard inequalities as in Proof of Proposition D4, we obtain the bounds

$$\begin{aligned}&\Big |{\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}_L}(q_i,q'_j)\big )- {\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}(q_i,q'_j)\big )\Big |\nonumber \\&\quad \le \,L^{-1}(r+1)^{\frac{r+1}{2}+1}\,C_{2k}^{r+1}\,\frac{1}{(1+p^2)^k(1+q^2)^k} \prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^{2k}} \end{aligned}$$

(D.32)

and, finally, the estimate

$$\begin{aligned} \big |\widehat{\mathcal{R}_L}(p,q)-\widehat{\mathcal{R}}(p,q)\big |\,\le \, \frac{L^{-1}C_k}{(1+p^2)^k(1+q^2)^k}\, \end{aligned}$$

(D.33)

for some $\,L$-independent constants $\,C_k$. This proves that $\,\mathcal{R}_L\,$ converge to $\,\mathcal{R}\,$ with speed $\,L^{-1}\,$ as operators of fast-decay type.

$\quad \square$

Proposition D8

Let, as in Proposition D5, $\,D_L\,$ be operators on $\,L^2(S^1_L)\,$ of fast-decay type and let $\,\delta D_L\,$ be similar operators converging with speed $\,L^{-1}\,$ to zero. Suppose that $\,D_L\,$ are the L-periodization of operators $\,\mathcal{D}_L\,$ on $\,L^2({\mathbb R})\,$ of fast-decay type satisfying uniform in $\,L\,$ fast-decay bounds. Assume additionally that there exists $\,L_0>0\,$ such that the Fredholm determinants $\,\det (I+D_L)\,$ are bounded away from zero uniformly in $\,L\le L_0$.  Then for $\,L\,$ large enough the Fredholm operators $\,I+\widetilde{D}_L\,$ for $\,\widetilde{D}_L=D_L+\delta D_L\,$ are invertible and the operators $\,\widetilde{R}_L=I-(I+\widetilde{D}_L)^{-1}\,$ are of fast-decay type. Besides there exist operators $\,\widetilde{\mathcal{R}}_L\,$ on $\,L^2({\mathbb R})\,$ of fast-decay type satisfying uniform in $\,L\,$ fast-decay bounds and such that $\,\widetilde{R}_L\,$ are their L-periodization.

Proof

From Proposition D5 it follows that $\,\det (I+\widetilde{D}_L)\,$ are bounded away from zero for $\,L\,$ large enough so that $\,I+\widetilde{D}_L\,$ are invertible. Let $\,\delta \mathcal{D}_L\,$ be the operators on $\,L^2({\mathbb R})\,$ of fast-decay type converging to zero with speed $\,L^{-1}\,$ and such that $\,\delta D_L\,$ are their L-periodization. Set $\,\widetilde{\mathcal{D}}_L=\mathcal{D}_L+\delta \mathcal{D}_L$.  The matrix elements of $\,\widetilde{R}_L\,$ are then given by the Fredholm series [27]

$$\begin{aligned} (\widetilde{R}_L)_{m,n}= & {} \frac{1}{\det (I+\widetilde{D}_L)}\, \sum \limits _{r=0}^\infty \,\frac{_1}{r!}\sum \limits _{(n_1,\dots ,n_r)\in {\mathbb Z}_{\not =0}^{\ r}}{\det }_{(r+1)\times (r+1)}\big ((\widetilde{D}_L)_{n_i,n'_j}\big )\nonumber \\= & {} \frac{1}{\det (I+\widetilde{D}_L)}\,\sum \limits _{r=0}^\infty \, \frac{1}{{r!\,L^{r+1}}}\sum \limits _{(n_1,\dots ,n_r)\in {\mathbb Z}_{\not =0}^{\ r}}{\det }_{(r+1)\times (r+1)} \big (\widehat{\widetilde{\mathcal{D}}_L}(p_{n_i},p_{n'_j})\big )\, \nonumber \\ \end{aligned}$$

(D.34)

where $\,n_0=m$, $\,n_0'=n$, $\,n_i=n'_i\,$ for $\,i=1,\dots ,r$,  and the $2\mathrm{nd}$ equality follows from the fact that $\,\widetilde{D}_L\,$ are the L-periodization of $\,\widetilde{\mathcal{D}}_L$.  Let us now define an operator $\,\widetilde{\mathcal{R}}_L\,$ on $\,L^2({\mathbb R})\,$ with the momentum-space kernel

$$\begin{aligned} \widehat{\widetilde{\mathcal{R}}_L}(p,q)\,=\,\frac{1}{\det (I+\widetilde{D}_L)}\, \sum \limits _{r=0}^\infty \,\frac{1}{{r!\,L^{r}}} \sum \limits _{(n_1,\dots ,n_r)\in {\mathbb Z}_{\not =0}^{\ r}} {\det }_{(r+1)\times (r+1)}\big ((\widehat{\widetilde{\mathcal{D}}_L}(\mathrm{q}_i,\mathrm{q}'_j) \big )\,,\nonumber \\ \end{aligned}$$

(D.35)

where

$$\begin{aligned} \mathrm{q}_0=p\,,\qquad \mathrm{q}'_0=q\,,\qquad \mathrm{q}_i=p_{n_i}=\mathrm{q}_i'\quad \ \text {for}\quad \ i=1,\dots ,r\,. \end{aligned}$$

(D.36)

Clearly, $\,\widetilde{R}_L\,$ is the L-periodization of $\,\widetilde{\mathcal{R}}_L$.  Since operators $\,\widetilde{\mathcal{D}}_L\,$ satisfy uniform fast-decay bounds, we get from the Hadamard inequality the uniform estimate

$$\begin{aligned} \Big |{\det }_{(r+1)\times (r+1)}\big (\widehat{\widetilde{\mathcal{D}}_L} (\mathrm{q}_i,\mathrm{q}'_j)\big |\Big | \,\le \,(r+1)^{\frac{r+1}{2}}C_{2k}^{r+1}\frac{1}{(1+p^2)^k(1+q^2)^k} \prod \limits _{i=1}^r\frac{1}{(1+p_{n_i}^2)^{2k}}\nonumber \\ \end{aligned}$$

(D.37)

leading to the uniform fast-decay bounds

$$\begin{aligned} \big |\widehat{\widetilde{\mathcal{R}}_L}(p,q)\big |\,\le \, \frac{C_k}{(1+p^2)^k(1+q^2)^k}\,. \end{aligned}$$

(D.38)

$\square$

Proposition D9

If $\,\mathcal{D}\,$ is an operator on $\,L^2({\mathbb R})\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type and $\,\det (\mathcal{I}+\mathcal{D})\not =0\,$ then $\,\mathcal{R}=\mathcal{I}-(\mathcal{I}+\mathcal{D})^{-1}\,$ is also of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type.  If, moreover, operators $\,\mathcal{D}_L\,$ on $\,L^2({\mathbb R})\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type converge to $\,\mathcal{D}\,$ with speed $\,L^{-1}\,$ then the operators $\,\mathcal{R}_L=\mathcal{I}-(\mathcal{I}+\mathcal{D}_L)^{-1}$,  well defined and of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type for $\,L\,$ large enough,  converge as such to $\,\mathcal{R}\,$ with speed $\,L^{-1}$.

Proof

The momentum-space kernel of $\,\mathcal{R}\,$ is given by (D.27).  Since for $\,(q_i,q'_i)\,$ as in (D.28) with $\,q_i\not =0\not =q'_i$,

$$\begin{aligned} \partial _p^{\ell _1}\partial _q^{\ell _2}\,{\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}(q_i,q'_j)\big )\,=\,{\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_j)\big )\,, \end{aligned}$$

(D.39)

where

$$\begin{aligned}&\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_0,q'_0)=\partial _p^{\ell _1}\partial _q^{\ell _2} \widehat{\mathcal{D}}(p,q)\,,\nonumber \\&\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_0,q'_j)=\partial _p^{\ell _1} \widehat{\mathcal{D}}(p,q_j)\quad \ \quad \; \text {for}\quad \ j=1,\dots ,r\,,\nonumber \\&\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_0)=\partial _q^{\ell _2} \widehat{\mathcal{D}}(q_i,q)\quad \ \,\quad \,\text {for}\quad \ i=1,\dots ,r\,,\nonumber \\&\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_j)= \widehat{\mathcal{D}}(q_i,q_j)\text {for}\quad \ i,j=1,\dots ,r\,, \end{aligned}$$

(D.40)

we infer from the Hadamard inequalities that

$$\begin{aligned}&\big |{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_j)\big )\big |\nonumber \\&\quad \le \, (r+1)^{\frac{r+1}{2}}C_{\ell _1,\ell _2,2k}^{r+1}\, \frac{1}{(1+p^2)^k(1+q^2)^k}\prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^{2k}} \end{aligned}$$

(D.41)

for some constants $\,C_{\ell _1,\ell _2,2k}\,$ and the bounds

$$\begin{aligned} \big |\partial _p^{\ell _1}\partial _q^{\ell _2}\widehat{\mathcal{R}}(p,q)\big |\,\le \frac{C_{\ell _1,\ell _2,k}}{(1+p^2)^k(1+q^2)^k} \end{aligned}$$

(D.42)

for some new constants $\,C_{\ell _1,\ell _2,k}\,$ follow. The statement about the convergence of $\,\mathcal{R}_L\,$ to $\,\mathcal{R}\,$ with speed $\,L^{-1}\,$ is inferred similarly from the bound

$$\begin{aligned}&\big |{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(q_i,q'_j)\big )- {\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_j)\big )\big |\nonumber \\&\quad \le \,L^{-1}(r+1)^{\frac{r+1}{2}+1}C_{\ell _1,\ell _2,2k}^{r+1}\, \frac{1}{(1+p^2)^k(1+q^2)^k}\prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^{2k}}\,. \end{aligned}$$

(D.43)

$\square$

Proposition D10

Let $\,D_L\,$ be operators on $\,L^2_0(S^1_L)\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type converging with speed $\,L^{-1}\,$ to an operator $\,\mathcal{D}\,$ on $\,L^2({\mathbb R})\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type such that $\,\det (\mathcal{I}+\mathcal{D})\not =0$.  Then for $\,L\,$ large enough, $\,I+D_L\,$ are invertible Fredholm operators and $\,R_L=I-(I+D_L)^{-1}\,$ are operators on $\,L^2_0(S^1_L)\,$ of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type converging with speed $\,L^{-1}\,$ to $\,\mathcal{R}=\mathcal{I}-(\mathcal{I}+\mathcal{D})^{-1}$.

Proof

Let $\,\mathcal{D}_L\,$ be the operators of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type on $\,L^2({\mathbb R})\,$ converging with speed $\,L^{-1}\,$ to $\,\mathcal{D}\,$ and such that $\,D_L\,$ are the L-periodization of $\,\mathcal{D}_L$,  see Definitions 3 and 4 in Sect. 10.3. From Propositions D4 and D6, we infer that

$$\begin{aligned} \big |\det (I+D_L)-\det (\mathcal{I}+\mathcal{D})\big |\,\le \,L^{-1}C \end{aligned}$$

(D.44)

for some $\,C$.  It follows then [27] that for $\,L\,$ large enough the Fredholm operator $\,I+D_L\,$ is invertible and $\,R_L=I-(I+D_L)^{-1}\,$ has the matrix elements given by the Fredholm series as in (D.34) but without tilde and is the L-periodization of the operator $\,\mathcal{R}_L\,$ given by (D.35), again without tilde. Now for $\,p,q\not =0$,

$$\begin{aligned} \partial _p^{\ell _1}\partial _q^{\ell _2}\,{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}_L}(\mathrm{q}_i,\mathrm{q}'_j)\big )\,=\,{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_j)\big )\,, \end{aligned}$$

(D.45)

where

$$\begin{aligned}&\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_0,\mathrm{q}'_0)=\partial _p^{\ell _1} \partial _q^{\ell _2}\widehat{\mathcal{D}_L}(p,q)\,,\nonumber \\&\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_0,\mathrm{q}'_j)=\partial _p^{\ell _1} \widehat{\mathcal{D}_L}(p,p_{n_j})\quad \,\,\,\, \text {for}\quad \ j=1,\dots ,r\,,\nonumber \\&\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_0)=\partial _q^{\ell _2} \widehat{\mathcal{D}_L}(p_{n_i},q)\text {for}\quad \ i=1,\dots ,r\,,\nonumber \\&\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_j)= \widehat{\mathcal{D}_L}(p_{n_i},p_{n_j})\text {for}\quad \ i,j=1,\dots ,r \end{aligned}$$

(D.46)

and

$$\begin{aligned}&\big |{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_j)\big )\big |\nonumber \\&\quad \le \, (r+1)^{\frac{r+1}{2}}C_{\ell _1,\ell _2,2k}^{r+1}\, \frac{1}{(1+p^2)^k(1+q^2)^k}\prod \limits _{i=1}^r\frac{1}{(1+p_{n_i}^2)^{2k}} \end{aligned}$$

(D.47)

for some constants $\,C_{\ell _1,\ell _2,2k}\,$ implying the bounds

$$\begin{aligned} \big |\partial _p^{\ell _1}\partial _q^{\ell _2}\widehat{\mathcal{R}_L}(p,q)\big |\,\le \frac{C_{\ell _1,\ell _2,k}}{(1+p^2)^k(1+q^2)^k}\,. \end{aligned}$$

(D.48)

This proves that $\,\mathcal{R}_L$,  and hence also $\,R_L$,  are operators of the Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type.

It remains to prove that $\,\mathcal{R}_L\,$ converge to $\,\mathcal{R}\,$ with speed $\,L^{-1}\,$ as Schwartz-type operators. Because of (D.44), it is enough to estimate

$$\begin{aligned}&\bigg |\sum _{n_1,\dots ,n_r\in {\mathbb Z}_{\not =0}^{\ r}}\frac{1}{L^r}\,\partial _p^{\ell _1} \partial _q^{\ell _2}\,{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}_L}(\mathrm{q}_i,\mathrm{q}'_j)\big )\nonumber \\&\qquad -\int \limits _{{\mathbb R}^r} \frac{1}{(2\pi )^r}\,\partial _p^{\ell _1}\partial _q^{\ell _2}\,{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}}(q_i,q'_j)\big )\,dq_1\cdots dq_r\bigg |\nonumber \\&\quad =\bigg |\sum _{n_1,\dots ,n_r\in {\mathbb Z}_{\not =0}^{\ r}}\frac{1}{L^r}\,{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_j)\big )\nonumber \\&\qquad - \int _{{\mathbb R}^r}\frac{1}{(2\pi )^r}\,{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_j)\big )\,dq_1\cdots dq_r\bigg |\nonumber \\&\quad \le \sum _{n_1,\dots ,n_r\in {\mathbb Z}_{\not =0}^{\ r}}\frac{1}{L^r}\, \bigg |{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}_L}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_j)\big )- {\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_j)\big )\bigg |\nonumber \\&\qquad +\bigg |\sum _{n_1,\dots ,n_r\in {\mathbb Z}_{\not =0}^{\ r}}\frac{1}{L^r}\, {\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(\mathrm{q}_i, \mathrm{q}'_j)\big )\nonumber \\&\qquad -\int _{{\mathbb R}^r}\frac{1}{(2\pi )^r}\,{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_j)\big )\,dq_1\cdots dq_r\bigg |.\qquad \end{aligned}$$

(D.49)

The $1\mathrm{st}$ sum on the right-hand side is estimated as in Proof of Proposition D7 by

$$\begin{aligned}&L^{-1}(r+1)^{\frac{r+1}{2}+1}\,C_{\ell _1,\ell _2,2k}^{r+1}\,\frac{1}{(1+p^2)^k(1+q^2)^k} \sum _{n_1,\dots ,n_r\in {\mathbb Z}_{\not =0}^{\ r}}\frac{1}{L^r}\,\prod \limits _{i=1}^r\frac{1}{ (1+p_{n_i})^{2k}}\nonumber \\&\quad \le L^{-1}(r+1)^{\frac{r+1}{2}+1}\,\frac{C_{\ell _1,\ell _2,k}^{r+1}}{(1+p^2)^k(1+q^2)^k}\,, \end{aligned}$$

(D.50)

with some new $\,C_{\ell _1,\ell _2,k}$,  compare to (D.43).  For the $2\mathrm{nd}$ term on the right-hand side of (D.49), we use for $\,q_i\in \widehat{J}_{n_i}\,$ with $\,i=1,\dots ,r\,$ the bound

$$\begin{aligned}&\big |{\det }_{(r+1)\times (r+1)} \big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(\mathrm{q}_i,\mathrm{q}'_j)\big ) -{\det }_{(r+1)\times (r+1)}\big (\widehat{\mathcal{D}}^{\ell _1,\ell _2}(q_i,q'_j) \big )\big |\nonumber \\&\quad \le L^{-1}(r+1)^{\frac{r+1}{2}+1}\,C^{r+1}_{\ell _1\ell _2,2k}\,\frac{1}{(1+p^2)^k (1+q^2)^k}\prod \limits _{i=1}^r\frac{1}{(1+q_i^2)^{2k}} \end{aligned}$$

(D.51)

and for at least one $\,q_i\in \widehat{J}_0$,  we extract the factor $\,L^{-1}\,$ from the length of $\,\widehat{J}_0$,  similarly as in the proof of Proposition D6. Altogether, this gives for the $2\mathrm{nd}$ term on the right hand side of (D.49) a similar bound as that for the $1\mathrm{st}$ one and permits to conclude the proof. $\quad \square$

Appendix E

Proof of Lemma 2

The support properties of the momentum-space kernels $\,\widehat{\mathcal{D}}_i(p,q)\,$ of the operators in question are evident. Now on $\,{\mathbb R}_+\times {\mathbb R}_-$,

$$\begin{aligned} \widehat{\mathcal{D}}_1(p,q)= & {} q^{-1} \int \mathrm{e}^{\mathrm {i}px}\Big (\mathrm{e}^{-\mathrm {i}qg(x)}- \mathrm{e}^{-\mathrm {i}qx}\Big )\;dx\nonumber \\= & {} q^{-1} \int \mathrm{e}^{\mathrm {i}px}\Big (\int _0^1\partial _\sigma \,\mathrm{e}^{-\mathrm {i}q(x+\sigma (g(x)-x))}\,d\sigma \Big )\,dx \nonumber \\= & {} -\mathrm {i}\int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px-\mathrm {i}q(x+\sigma (g(x)-x))}\,(g(x)-x)\,dx \end{aligned}$$

(E.1)

so that

$$\begin{aligned}&\partial _{p}^{\ell _1}\partial _{q}^{\ell _2}\widehat{\mathcal{D}}_1(p,q)\nonumber \\&\quad =-\mathrm {i}\int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px-\mathrm {i}q(x+\sigma (g(x)-x))}\, \big (\mathrm {i}x\big )^{\ell _1}\,\big (-\mathrm {i}(x+\sigma (g(x)-x))\big )^{\ell _2} \,(g(x)-x)\,dx\nonumber \\&\quad =-\mathrm {i}\int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px-\mathrm {i}q(x+\sigma (g(x)-x))}\, d_1^n\Big (\big (\mathrm {i}x\big )^{\ell _1}\,\big (-\mathrm {i}(x+(g(x)-x))\big )^{\ell _2} \,(g(x)-x)\Big )\,dx \nonumber \\ \end{aligned}$$

(E.2)

for $\,n=0,1,\dots \,$ and

$$\begin{aligned} (d_1\mathcal{X})(x)\,=\,(\mathrm {i}\partial _x)\Big (\frac{_1}{^{p-q(1+\sigma (g'(x)-1))}}\mathcal{X}(x) \Big )\,. \end{aligned}$$

(E.3)

The last equality in (E.2) follows by the subsequent integration by parts over $\,x\,$ in which all the boundary terms vanish because of the compact support of $\,g(x)-x$. Since $\,|p-q(1+\sigma (g'(x)-1))|\ge |p|+\epsilon |q|\,$ if $\,p\,$ and $\,q\,$ have different signs for some $\,\epsilon >0\,$ independent of $\,x\,$ and $\,\sigma$,  it follows that

$$\begin{aligned} \big |\partial _{p}^{\ell _1}\partial _{q}^{\ell _2}\widehat{\mathcal{D}}_1(p,q)\big |\,\le \, \frac{C_{\ell _1,\ell _2,n}}{(|p|+|q|)^n} \end{aligned}$$

(E.4)

from which the claim of Lemma 2 follows for $\,\mathcal{D}_1$.  The claim for $\,\mathcal{D}_2\,$ follows the same way.  For $\,\mathcal{D}_3$,

$$\begin{aligned} \widehat{\mathcal{D}}_3(p,q)\,=\, -\mathrm {i}\,\mathrm{e}^{-\gamma _\# q} \int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px-\mathrm {i}q(x+\sigma (g(x)-x))}\,(g(x)-x)\,dx \end{aligned}$$

(E.5)

on $\,{\mathbb R}_+\times {\mathbb R}_+$,  where $\,\gamma _\#=\gamma _L\,$ or $\,\gamma _\#=\gamma \,$ so that

$$\begin{aligned}&\partial _{p}^{\ell _1}\partial _{q}^{\ell _2}\widehat{\mathcal{D}}_3(p,q)\nonumber \\&\quad =-\mathrm {i}\,\mathrm{e}^{-\gamma _\# q} \int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px-\mathrm {i}q(x+\sigma (g(x)-x))}\, \big (\mathrm {i}x\big )^{\ell _1}\, \big (-\gamma _\#-\mathrm {i}(x+\sigma (g(x)-x))\big )^{\ell _2}\nonumber \\&\qquad \times (g(x)-x)\,dx\,. \end{aligned}$$

(E.6)

It follows that

$$\begin{aligned}&(1+p^2)^k(1+q^2)^k\,\big |\partial _{p}^{\ell _1}\partial _{q}^{\ell _2}\widehat{\mathcal{D}}_3(p,q)\big |\nonumber \\&\quad =\mathrm{e}^{-\gamma _\# q}\;(1+q^2)^k\,\Big |\int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px} \big (1-\partial _x^2)^{k}\Big ( \big (\mathrm {i}x\big )^{\ell _1}\big (-\gamma _\#-\mathrm {i}(x+\sigma (g(x)-x))\big )^{\ell _2}\nonumber \\&\qquad \times (g(x)-x)\,\mathrm{e}^{-\mathrm {i}q(x+\sigma (g(x)-x))}\Big )dx\Big |\nonumber \\&\quad \le \,c_{\ell _1,\ell _2,k}\;\mathrm{e}^{-\gamma _\# q}(1+q^2)^{2k}\,\le \,C_{\ell _1,\ell _2,k} \end{aligned}$$

(E.7)

which gives the claim of Lemma for $\,\mathcal{D}_3$. For $\,\mathcal{D}_4$,

$$\begin{aligned} \widehat{\mathcal{D}}_4(p,q)\,=\, -\mathrm {i}\,\mathrm{e}^{\gamma _\# p} \int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px-\mathrm {i}q(x+\sigma (g(x)-x))}\,(g(x)-x)\,dx \end{aligned}$$

(E.8)

on $\,{\mathbb R}_-\times {\mathbb R}_-\,$ so that

$$\begin{aligned}&\partial _{p}^{\ell _1}\partial _{q}^{\ell _2}\widehat{\mathcal{D}}_4(p,q)\nonumber \\&\quad =\,-\mathrm {i}\,\mathrm{e}^{\gamma _\# p} \int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}px-\mathrm {i}q(x+\sigma (g(x)-x))}\, \big (\gamma _\#+\mathrm {i}x\big )^{\ell _1}\, \big (-\mathrm {i}(x+\sigma (g(x)-x))\big )^{\ell _2}\nonumber \\&\qquad \times (g(x)-x)\,dx\,.\qquad \end{aligned}$$

(E.9)

Hence

$$\begin{aligned}&(1+p^2)^k(1+q^2)^k\,\big |\partial _{p}^{\ell _1}\partial _{q}^{\ell _2} \widehat{\mathcal{D}}_4(p_1,p_2)\big |\nonumber \\&\quad =\mathrm{e}^{\gamma _\# p}\,(1+p^2)^k \Big |\int _0^1d\sigma \int \mathrm{e}^{-\mathrm {i}q(x+\sigma (g(x)-x))}(1+d_2^2)^k\Big (\mathrm{e}^{\mathrm {i}px} \big (\gamma _\#+\mathrm {i}x\big )^{\ell _1}\nonumber \\&\qquad \times \big (-\mathrm {i}(x+\sigma (g(x)-x))\big )^{\ell _2} \,(g(x)-x)\Big )\,dx\Big |, \end{aligned}$$

(E.10)

where

$$\begin{aligned} (d_2\mathcal{X})(x)=-\mathrm {i}\partial _x\Big (\frac{_1}{^{1+\sigma (g'(x)-1)}}\mathcal{X}(x)\Big ). \end{aligned}$$

(E.11)

It follows that

$$\begin{aligned} (1+p^2)^k(1+q^2)^k\,\big |\partial _{p}^{\ell _1}\partial _{q}^{\ell _2} \widehat{\mathcal{D}}_4(p,q)\big |\,\le \,c_{\ell _1,\ell _2,k}\,\mathrm{e}^{\gamma _\# p}\,(1+p^2)^{2k} \,\le \,C_{\ell _1,\ell _2,k} \end{aligned}$$

(E.12)

which proves the claim of Lemma for $\,\mathcal{D}_4$. For $\,\mathcal{D}_5$,

$$\begin{aligned} \widehat{\mathcal{D}_5}(p,q)=\frac{_1}{^{2\pi }}\int _0^\infty \widehat{a}_+(p,r)\, \widehat{a}_-(r,q)\,r\,dr \end{aligned}$$

(E.13)

on $\,{\mathbb R}_+\times {\mathbb R}_-$,  where

$$\begin{aligned} \widehat{a}_\pm (p,q)=-\mathrm {i}\int _0^1d\sigma \int \mathrm{e}^{\mathrm {i}rx-\mathrm {i}q(y+\sigma (g^{\pm 1}(x)-x))}(g^{\pm 1}(x)-x)\,dx \,. \end{aligned}$$

(E.14)

Estimating as for $\,\mathcal{D}_1\,$ and $\,\mathcal{D}_3$,  we infer that for $\,pq<0$,

$$\begin{aligned} \big |\partial _p^\ell \partial _q^{\ell _2}\widehat{a}_\pm (p,q)\big | \,\le \,\frac{c_{\ell ,\ell _2,k}}{(1+p^2)^k(1+q^2)^k}\,, \end{aligned}$$

(E.15)

and for $\,pq>0$,

$$\begin{aligned} \big |\partial _p^{\ell _1}\partial _q^\ell \widehat{a}_\pm (p,q)\big | \,\le \,\frac{c_{\ell _1,\ell ,k}(1+q^2)^k}{(1+p^2)^k}\,. \end{aligned}$$

(E.16)

The above estimates with $\,\ell =0\,$ imply that

$$\begin{aligned} \big |\partial _p^{\ell _1}\partial _q^{\ell _2} \widehat{\mathcal{D}_5}(p,q)\big |\,\le \,\frac{C_{\ell _1,\ell _2,k}}{(1+p^2)^k(1+q^2)^k} \end{aligned}$$

(E.17)

as claimed. For the later use, let us observe that if we consider operator $\,\mathcal{D}_{5,L}\,$ with the momentum-space kernel

$$\begin{aligned} \widehat{\mathcal{D}_{5,L}}(p,q)=\frac{_1}{^L} \sum \limits _{n=1}^\infty \widehat{a}_+(p,p_n)\,\widehat{a}_-(p_n,q)\,p_n \end{aligned}$$

(E.18)

for $\,p_n=\frac{2\pi n}{L}\,$ then the estimates (E.15) and (E.16) with $\,\ell =0,1\,$ show that

$$\begin{aligned} \big |\partial _p^{\ell _1}\partial _q^{\ell _2}\widehat{\mathcal{D}_{5,L}}(p,q) -\partial _p^{\ell _1}\partial _q^{\ell _2}\widehat{\mathcal{D}_5}(p,q)\big |\,\le \,\frac{L^{-1}C_{\ell _1,\ell _2,k}}{(1+p^2)^k(1+q^2)^k}\,, \end{aligned}$$

(E.19)

i.e.  that $\,\mathcal{D}_{5,L}\,$ converge to $\,\mathcal{D}_5\,$ with speed $\,L^{-1}\,$ as operators of Schwartz ${\mathbb R}_+\times {\mathbb R}_-$ type. Similarly we prove that $\,\mathcal{D}_6\,$ is of Schwartz ${\mathbb R}_-\times {\mathbb R}_+$ type and that its analogous modifications $\,\mathcal{D}_{6,L}\,$ converge with speed $\,L^{-1}\,$ to $\,\mathcal{D}_6$. $\quad \square$

If $\,g_L\,$ is a family of diffeomorphisms of $\,{\mathbb R}\,$ such that $\,g_L(x)=g(x)=x\,$ outside an $\,L$-independent bounded set and if for $\,\ell =0,1,\dots ,\,$

$$\begin{aligned} |\partial _x^\ell g_L(x)-\partial _x^\ell g(x)|\le L^{-1}C_\ell \end{aligned}$$

(E.20)

uniformly in $\,x\,$ then a small modification of the above proof shows that the operators $\,\mathcal{D}_{i,L}\,$ for $\,i=1,\dots ,6\,$ obtained from $\,\mathcal{D}_i\,$ by replacement of $\,g\,$ by $\,g_L\,$ and, for $\,i=5,6$, by cumulating this change with the one discussed above, converge as operators of Schwartz type to $\,\mathcal{D}_i\,$ with speed $\,L^{-1}$.

Appendix F

Proof of Lemma 3

$\mathcal{I}+\varSigma ^\pm \,$ is a Fredholm operator in $\,L^2({\mathbb R})\,$ since $\,\varSigma ^\pm \,$ is Hilbert–Schmidt as it is of Schwartz ${\mathbb R}_{\not =0}\times {\mathbb R}_{\not =0}$ type. We have then to show that $\,\mathcal{I}+\varSigma ^\pm \,$ has a trivial kernel,  i.e. that, if we assume that for some $\,\mathcal{Z}^\pm \in L^2({\mathbb R})\,$$\,(\mathcal{I}+\varSigma ^\pm )\mathcal{Z}^\pm \,$ vanishes, then it follows that $\,\mathcal{Z}^\pm =0$.  The assumption implies that for $\,\ell ,k=0,1,\dots \,$ and $\,p\not =0$,

$$\begin{aligned} \big |\partial _p^\ell \widehat{\mathcal{Z}^\pm }(p)\big |\,\le \,\frac{C_{\ell ,k}}{(1+p^2)^k}\,. \end{aligned}$$

(F.1)

in virtue of the Schwartz-type property of $\,\varSigma ^\pm$. If we write $\,\mathcal{Z}^\pm =(\mathcal{I}-\mathcal{K}^0)\mathcal{P}^{-1}\mathrm {i}\mathcal{Y}'_1\,$ then $\,\mathcal{Y}'_1\,$ also satisfies the last bound (with new constants). It follows that $\,\mathcal{Y}_1'(x)\,$ is smooth and is bounded by $\,C(1+|x|)^{-1}\,$ for some $\,C\,$ (as $\,\widehat{\mathcal{Y}_1'}(p)\,$ may have a jump at $\,p=0$).  Besides, from the relation

$$\begin{aligned} \varSigma ^\pm =-(\mathcal{K}-\mathcal{K}^0)(\mathcal{I}-\mathcal{K}^0)^{-1}\,, \end{aligned}$$

(F.2)

where $\,\mathcal{K}\,$ is given by (8.53) and (8.54) or (8.49)–(8.51) for $\,g=g^\pm _{s,t}$,  it follows that

$$\begin{aligned} (\mathcal{I}-\mathcal{K})\mathcal{P}^{-1}\mathcal{Y}'_1=0\,. \end{aligned}$$

(F.3)

Let $\,\mathcal{Y}_1\,$ be a function whose derivative is equal to $\,\mathcal{Y}_1'$. $\,\mathcal{Y}_1\,$ is smooth and determined up to a constant and $\,|\mathcal{Y}_1(x)|\le C\ln (2+|x|)\,$ for some $\,C$.  Then

$$\begin{aligned} (\mathcal{I}-\mathcal{K})\mathcal{Y}_1=0\,, \end{aligned}$$

(F.4)

where $\,\mathcal{K}_{11}\mathcal{Y}_1\,$ and $\,(\mathcal{K}_{12}+\mathcal{K}_{21})\mathcal{Y}_1\,$ are well defined and $\,\mathcal{K}_{11}1=0,\ \,(\mathcal{K}_{12}+\mathcal{K}_{21})1=1\,$ so that the zero mode equation (F.4) is solved for all choices of $\,\mathcal{Y}_1$.  Let us define a holomorphic function $\,\mathcal{Y}(z)\,$ on the interior of $\,\mathcal{B}_{g,\gamma }\,$ by

$$\begin{aligned} \mathcal{Y}(z)=\frac{_1}{^{2\pi \mathrm {i}}}\int \Big (\frac{g'(y)}{g(y)-\mathrm {i}\gamma -z}- \frac{1}{y-z}\Big )\,\mathcal{Y}_1(y)\,dy\,. \end{aligned}$$

(F.5)

where $\,g=g^\pm _{s,t}$.  Note that the integral converges since $\,g(y)=g^\pm _{s,t}(y)=y\,$ for large $\,|y|$.  Besides, if we add a constant to $\,\mathcal{Y}_1\,$ then the same constant is added to $\,\mathcal{Y}$.  A straightforward estimation shows that for $\,z=z_1+\mathrm {i}z_2$,

$$\begin{aligned} |\mathcal{Y}(z)|\,\le \,C\ln (2+|z_1|) \end{aligned}$$

(F.6)

for some $\,C$.  We shall show below that the boundary values of the function $\,\mathcal{Y}\,$ satisfy the relation

$$\begin{aligned} \mathcal{Y}\circ p_i=\mathcal{Y}_1+c \end{aligned}$$

(F.7)

for $\,p_i\,$ given by (8.25) and for the same constant $\,c\,$ for $\,i=1,2$.  In the variable $\,u=\mathrm{e}^{\frac{2\pi }{\gamma }z}\,$ (keeping the same notation for the function), $\,|\mathcal{Y}(u)|\le C\ln \ln (|u|+|u|^{-1})\,$ in virtue of (F.6) and a similar estimate holds in the variable $\,w=\mathcal{W}(u)$,  where $\,\mathcal{W}\,$ is the map discussed in Sect. 8.2 with the properties (8.34) and (8.36). Besides, $\,\mathcal{Y}\,$ is analytic in the complex variable $\,w\,$ everywhere except at zero and at infinity because of (F.7). Let

$$\begin{aligned} \mathcal{Y}_+(w)=\frac{_1}{^{2\pi \mathrm {i}}}\oint \limits _{|w'|=R}\frac{\mathcal{Y}(w')}{w'-w}\,dw' \end{aligned}$$

(F.8)

for any $\,R>|w|$. $\,\mathcal{Y}_+\,$ is an entire function on $\,\mathbb C$.  Since

$$\begin{aligned} \mathcal{Y}'_+(w)=\frac{_1}{^{2\pi \mathrm {i}}}\oint \limits _{|w'|=R}\frac{\mathcal{Y}(w')}{(w'-w)^2}\,dw'\,, \end{aligned}$$

(F.9)

taking $\,R\rightarrow \infty$,  we infer from the a priori bound

$$\begin{aligned} |\mathcal{Y}(w)|\,\le \,C\ln \ln (|w|+|w|^{-1}) \end{aligned}$$

(F.10)

that $\,\mathcal{Y}_+'=0\,$ and $\,\mathcal{Y}_+=\mathrm{const}$.  Similarly, let

$$\begin{aligned} \mathcal{Y}_-(w)=\frac{_1}{^{2\pi \mathrm {i}}}w^{-1} \oint \limits _{|w'|=R}\frac{\mathcal{Y}(w'^{-1})}{w'(w'-w^{-1})}\,dw' \end{aligned}$$

(F.11)

for any $\,R>|w|^{-1}$. $\,\mathcal{Y}_-\,$ is holomorphic on $\,\mathbb C^\times \,$ and vanishes at infinity. Taking $\,R\rightarrow \infty$,  we infer from the a priori bound (F.10) that $\,\mathcal{Y}_-=0$.  But $\,\mathcal{Y}=\mathcal{Y}_-+\mathcal{Y}_+\,$ as $\,\mathcal{Y}_+\,$ is given by the part of the Laurent series of $\,\mathcal{Y}\,$ with nonnegative powers and $\,\mathcal{Y}_-\,$ by the one with negative ones. Hence $\,\mathcal{Y}=\mathrm{const}.\,$ and, consequently, $\,\mathcal{Y}_1=\mathrm{const}.$, $\,\mathcal{Y}'_1=0\,$ and $\,\mathcal{Z}^\pm =0$.

It remains to show (F.7). To this end, let us first consider the derivative of function $\,\mathcal{Y}$,

$$\begin{aligned} \mathcal{Y}'(z)\,=\,\frac{_1}{^{2\pi \mathrm {i}}} \int \Big (\frac{_1}{g(y)-\mathrm {i}\gamma -z}-\frac{1}{y-z}\Big )\mathcal{Y}'_1(y)\,dy\,. \end{aligned}$$

(F.12)

Due to the decay of $\,\mathcal{Y}'_1$,  one has an a priori bound

$$\begin{aligned} |\mathcal{Y}'(z)|\,\le \,C(1+|z_1|)^{-1} \end{aligned}$$

(F.13)

for some $\,C$.  For the boundary values of $\,\mathcal{Y}'$,  we obtain the equations

$$\begin{aligned} \mathcal{Y}'\circ p_1=\mathcal{E}_-({g'}^{-1}\mathcal{Y}'_1)+\mathcal{K}_{11}({g'}^{-1}\mathcal{Y}'_1) +\mathcal{K}_{12}\mathcal{Y}'_1\,,\qquad \mathcal{Y}'\circ p_2=\mathcal{E}_+\mathcal{Y}'_1+\mathcal{K}_{21}({g'}^{-1}\mathcal{Y}'_1) \nonumber \\ \end{aligned}$$

(F.14)

similarly as in (8.44), (8.45) and (8.48),  except that we do not know yet that $\,\mathcal{Y}'\circ p_1={g'}^{-1}\mathcal{Y}'_1\,$ and $\,\mathcal{Y}'\circ p_2=\mathcal{Y}'_1\,$ and we would like to show it. Let us now consider for $\,\mathrm{Im}(z)>0\,$ and $\,\mathrm{Im}(z)<-\gamma \,$ the holomorphic function

$$\begin{aligned} \mathcal{U}(z) \,=\,\mathrm{sgn}(\mathrm{Im}(z))\frac{_1}{^{2\pi \mathrm {i}}} \int \Big (\frac{1}{g(y)-\mathrm {i}\gamma -z} -\frac{1}{y-z}\Big )\mathcal{Y}'_1(y)\,dy\,, \end{aligned}$$

(F.15)

with the boundary values

$$\begin{aligned} \mathcal{U}\circ p_1 =\mathcal{E}_+({g'}^{-1}\mathcal{Y}'_1)-\mathcal{K}_{11}({g'}^{-1}\mathcal{Y}'_1) -\mathcal{K}_{12}\mathcal{Y}'_1\,,\quad \mathcal{U}\circ p_2=-\mathcal{E}_-\mathcal{Y}'_1+\mathcal{K}_{21}({g'}^{-1}\mathcal{Y}'_1)\nonumber \\ \end{aligned}$$

(F.16)

such that

$$\begin{aligned}&\mathcal{U}\circ p_1-\mathcal{U}\circ p_2 \,=\,\mathcal{E}_+({g'}^{-1}\mathcal{Y}'_1)-\mathcal{K}_{11}({g'}^{-1}\mathcal{Y}'_1)-\mathcal{K}_{12}\mathcal{Y}'_1 +\mathcal{E}_-\mathcal{Y}'_1-\mathcal{K}_{21}({g'}^{-1}\mathcal{Y}'_1)\,. \nonumber \\ \end{aligned}$$

(F.17)

Differentiating (F.4) with the use of the relations

$$\begin{aligned} \partial \mathcal{K}_{11}=g'\mathcal{K}_{11}{g'}^{-1}\partial -\mathcal{E}_-\partial +g'\mathcal{E}_-{g'}^{-1}\partial \,,\quad \partial \mathcal{K}_{12}=g' \mathcal{K}_{12}\partial \,,\quad \partial \mathcal{K}_{21}=\mathcal{K}_{21}{g'}^{-1}\partial \,,\nonumber \\ \end{aligned}$$

(F.18)

we obtain after some algebra the identity

$$\begin{aligned} g'\Big (\mathcal{E}_+({g'}^{-1}\mathcal{Y}'_1)-\mathcal{K}_{11}({g'}^{-1}\mathcal{Y}'_1)-\mathcal{K}_{12}\mathcal{Y}'_1 \Big )+\mathcal{E}_-\mathcal{Y}'_1-\mathcal{K}_{21}({g'}^{-1}\mathcal{Y}'_1)\,=\,0 \end{aligned}$$

(F.19)

that substituted to (F.17) yields the equality

$$\begin{aligned} \mathcal{U}\circ p_1\,=\,\mathcal{U}\circ p_2 \,+\,\big (1-{g'}^{-1}\big )\Big (\mathcal{E}_-\mathcal{Y}'_1-\mathcal{K}_{21}({g'}^{-1}\mathcal{Y}'_1)\Big )\,=\, {g'}^{-1}\mathcal{U}\circ p_2. \end{aligned}$$

(F.20)

From (F.16) and (8.54) it follows that

$$\begin{aligned} \mathcal{E}_-\mathcal{G}\,\mathcal{U}\circ p_1=0\,,\qquad \mathcal{E}_+\,\mathcal{U}\circ p_2=0\,. \end{aligned}$$

(F.21)

Thus

$$\begin{aligned} (\mathcal{G}{g'}^{-1})_{--}\,\mathcal{U}\circ p_2=0. \end{aligned}$$

(F.22)

But $\,(\mathcal{G}{g'}^{-1})_{--}$ is the hermitian adjoint of $\,(\mathcal{G}^{-1})_{--}\,$ and the operator $\,(\mathcal{G}^{-1})_{--}\,$ is invertible on $\,\mathcal{E}_-L^2({\mathbb R})$. The latter fact is well known but let us digress to indicate how it is proven. First one shows that $\,(\mathcal{G}^{-1})_{--}\,$ is injective since if $\,(\mathcal{G}^{-1})_{--}\mathcal{X}=0\,$ for $\,\mathcal{X}\in \mathcal{E}_- L^2({\mathbb R})\,$ then $\,\mathcal{X}\,$ is a boundary value of a holomorphic function on the upper half-plane that vanishes at infinity and $\,\mathcal{X}\circ g\,$ is a boundary value of a holomorphic function on the lower half-plane that also vanishes at infinity. Such functions define a holomorphic function vanishing at one point on the Riemann sphere welded from the two compactified half-planes using the diffeomorphism $\,g\,$ so that $\,\mathcal{X}\,$ must vanish. Similarly, one shows that $\,\mathcal{G}_{--}\,$ is injective. But

$$\begin{aligned} (\mathcal{G}^{-1})_{--}\mathcal{G}_{--}=\mathcal{E}_--(\mathcal{G}^{-1})_{-+}\mathcal{G}_{+-} \end{aligned}$$

(F.23)

and the right hand side is a Fredholm operator of index zero on $\,\mathcal{E}_-L^2({\mathbb R})\,$ because $\,(\mathcal{G}^{-1})_{-+}\,$ and $\,\mathcal{G}_{+-}\,$ are Hilbert–Schmidt by Lemma 2 of Sect. 10.3 and it has no kernel by the injectivity of the left-hand side. Hence $\,(\mathcal{G}^{-1})_{--}\mathcal{G}_{--}\,$ is invertible on $\,\mathcal{E}_-L^2({\mathbb R})\,$ and so are $\,(\mathcal{G}^{-1})_{--}\,$ and its hermitian adjoint $\,(\mathcal{G}{g'}^{-1})_{--}$. As a consequence, the relations (F.22) and (F.20) imply that $\,\mathcal{U}\circ p_i=0$.  We are almost done since the latter relations together with (F.16) and (F.14) imply that

$$\begin{aligned} \mathcal{Y}'\circ p_1={g'}^{-1}\mathcal{Y}'_1\,,\qquad \mathcal{Y}'\circ p_2=\mathcal{Y}'_1 \end{aligned}$$

(F.24)

so that

$$\begin{aligned} \mathcal{Y}\circ p_1=\mathcal{Y}_1+c_1\,,\qquad \mathcal{Y}\circ p_2=\mathcal{Y}_1+c_2 \end{aligned}$$

(F.25)

for some constants $\,c_i$.  But, for $\,x\,$ sufficiently large,

$$\begin{aligned} c_1-c_2=\mathcal{Y}(x-\mathrm {i}\gamma )-\mathcal{Y}(x)=\mathrm {i}\int _{-\gamma }^0\mathcal{Y}'(x+\mathrm {i}y)\,dy \end{aligned}$$

(F.26)

which is bounded by $\,O\big (\frac{1}{1+|x|}\big )\,$ in virtue of (F.13) so that $\,c_1-c_2\,$ must vanish. This establishes (F.7) completing the proof of Lemma 3. $\quad \square$

Appendix G

Proof of Lemma 8

For $\,\mathcal{D}^\pm _{7,L}\,$ we take the operator with the momentum-space kernel

$$\begin{aligned} \widehat{\mathcal{D}^\pm _{7,L}}(p,q)=\frac{_1}{^L}\sum \limits _{n=1}^{\infty } \widehat{a}^\pm _+(p,-p_n)\,\widehat{a}_-^\pm (-p_n,q)\,(-p_n)\,\mathrm{e}^{\pm \mathrm {i}p_nM_L} \end{aligned}$$

(G.1)

on ${\mathbb R}_+\times {\mathbb R}_+$,  where $\,\widehat{a}^\pm _\pm (p,q)\,$ and are given by (E.14) for $\,g=g^\pm _{s,t,L}\,$ and they satisfy the estimates (E.15) and (E.16) uniformly in $\,L$.  Note that these bounds imply that

$$\begin{aligned} \big |\partial _p^{\ell _1}\partial _r^\ell \partial _q^{\ell _2}\big ( \widehat{a}^\pm _+(p,r)\,\widehat{a}^\pm _-(r,q)\,r\big )|\,\le \, \frac{C_{\ell _1,\ell ,\ell _2,k}}{(1+p^2)^k(1+r^2)^k(1+q^2)^k} \end{aligned}$$

(G.2)

for non-zero $\,p,r,q\,$ not all of the same sign.  Using the summation by parts formula (10.81) in which we set $\,u_n=\widehat{a}^\pm _+(p,-p_n)\,\widehat{a}^\pm _-(-p_n,q) \,(-p_n)\,$ and $\,v_n=\mathrm{e}^{\pm \mathrm {i}p_nM_L}$,  we infer that for $\,L\,$ sufficiently large,

$$\begin{aligned}&\widehat{\mathcal{D}^\pm _{7,L}}(p,q)=\frac{_1}{^L}\bigg ( \widehat{a}^\pm _+(p,-p_1)\,\widehat{a}^\pm _-(-p_1,q) \,p_1\nonumber \\&\qquad +\sum \limits _{n=1}^\infty \Big (\widehat{a}^\pm _+(p,-p_{n+1})\, \widehat{a}^\pm _-(-p_{n+1},q) \,p_{n+1}\nonumber \\&\qquad -\widehat{a}^\pm _+(p,-p_n)\,\widehat{a}^\pm _-(-p_n,q) \,p_n\Big )\,\frac{1-\mathrm{e}^{\pm \mathrm {i}p_{n+1}M_L}}{1-\mathrm{e}^{\pm \mathrm {i}p_1M_L}}\bigg ), \end{aligned}$$

(G.3)

where we used the facts that by (G.2), $\,u_m\mathop {\longrightarrow }\limits _{m\rightarrow \infty }0$,  and that $\,s_m\,$ are bounded uniformly in $\,L\,$ sufficiently large, see (8.23). The bound (G.2) also implies that

$$\begin{aligned}&\Big |\Big (\widehat{a}^\pm _+(p,-p_{n+1})\, \widehat{a}^\pm _-(-p_{n+1},q) \,p_{n+1}\,-\,\widehat{a}^\pm _+(p,-p_n)\,\widehat{a}^\pm _-(-p_n,q) \,p_n\Big )\Big |\nonumber \\&\quad \le \frac{L^{-1}C_k}{(1+p^2)(1+p_n^2)(1+q^2)^k} \end{aligned}$$

(G.4)

which used on the right hand side of (G.3) gives the estimate

$$\begin{aligned} \big |\widehat{\mathcal{D}^\pm _{7,L}}(p,q)\big |\,\le \, \frac{L^{-1}C_k}{(1+p^2)^k(1+q^2)^k} \end{aligned}$$

(G.5)

showing that $\,\mathcal{D}^\pm _{7,L}\,$ converge to zero with speed $\,L^{-1}\,$ as operators of fast-decay type. The case of operators $\,\widehat{\mathcal{D}^\pm _{i,L}}\,$ with $\,i=8,\cdots ,10\,$ is treated the same way using again the summation by parts formula (10.81) and the estimates (G.2). $\quad \square$