CGO-Faddeev approach for complex conductivities with regular jumps in two dimensions

We denote by $n(z)=\left(n_x(z),n_y(z)\right)$ the unit outer normal vector in $\Gamma$ and on the complex plane by $\nu(z)=n_x(z)+{\rm i}n_y(z)$. During the paper we consider two orientations for the contour $\Gamma$: positively oriented $\Gamma^+$ (curve interior $\mathcal{D}$ is to the left) and negatively oriented $\Gamma^-$ (curve interior is to the right).

Lemma 2.1. The transmission condition (2) implies the following condition to the Dirac equation on $\Gamma$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{7JunB} \left( \begin{array}{@{}c@{}} \phi_1^+ - \phi_1^- \nonumber \\ \phi_2^+ - \phi_2^- \nonumber \\ \end{array} \right) = \frac{1}{2}\left( \begin{array}{@{}cc@{}} \alpha+\frac{1}{\alpha}-2 & (\alpha-\frac{1}{\alpha})\bar{\nu}^2 \nonumber \\ (\alpha-\frac{1}{\alpha})\nu^2 & \alpha+\frac{1}{\alpha}-2 \nonumber \\ \end{array} \right) \left( \begin{array}{@{}c@{}} \phi_1^- \nonumber \\ \phi_2^- \nonumber \\ \end{array} \right) \nonumber \end{align} \tag{ 6 }$$

where $\alpha=\sqrt{\frac{\gamma^-}{\gamma^+}}$.

Proof. Let $l(z)=(-n_y(z),n_x((z))$ be a unit tangential vector to $\Gamma$. From the first equation of (2) follows for the tangential derivative that $\frac{\partial}{\partial l} (u^+(z)-u^-(z))=0$ and, therefore,

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sqrt{\gamma^+} u_l^+ - \sqrt{\gamma^-} u_l^- =u_l^- \sqrt{\gamma^-} (\frac{1}{\alpha}-1), \nonumber \end{align*}$$

where $u_l=\frac{\partial u}{\partial l}$. Moreover, during this proof and to simplify the computations we denote the normal derivative as $u_n=\frac{\partial u}{\partial \nu}$. From the second equation of (2) we get $u_n^+ =\frac{\gamma^-}{\gamma^+} u_n^-$, where $u_n^{\pm}$ denotes the normal derivative of $u^{\pm}$, so that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sqrt{\gamma^+} u_n^+ - \sqrt{\gamma^-} u_n^-=\sqrt{\gamma^-} u_n^- (\alpha-1). \nonumber \end{align*}$$

Note that we have now

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\partial}u = \frac{1}{2} (\bar{\nu} u_n - {\rm i}\bar{\nu} u_l), \label{8} \nonumber \end{align} \tag{ 7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \bar{\partial}u = \frac{1}{2} (\nu u_n + {\rm i}\nu u_l), \label{9} \nonumber \end{align} \tag{ 8 }$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \phi_1^+-\phi_1^-= \sqrt{\gamma^+} {\partial}u^+ - \sqrt{\gamma^-} {\partial}u^-= \left(\frac{1}{\alpha}-1\right) u_l^- \sqrt{\gamma^-} \frac{1}{2}(-{\rm i}\bar{\nu}) + ({\alpha}-1) u_n^- \sqrt{\gamma^-} \frac{1}{2}\bar{\nu}, \nonumber \\ \phi_2^+-\phi_2^-= \sqrt{\gamma^+} \bar{\partial}u^+ - \sqrt{\gamma^-} \bar{\partial}u^-= \left(\frac{1}{\alpha}-1\right) u_l^- \sqrt{\gamma^-} \frac{1}{2}({\rm i}\nu) + \left({\alpha}-1\right) u_n^- \sqrt{\gamma^-} \frac{1}{2}\nu. \nonumber \end{align*}$$

These relations take the matricial form

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left( \begin{array}{@{}cc@{}} \phi_1^+-\phi_1^- \nonumber \\ \phi_2^+-\phi_2^- \nonumber \\ \end{array} \right) = \frac{1}{2}\left( \begin{array}{@{}cc@{}} (\alpha-1)\bar{\nu} & (\frac{1}{\alpha}-1) (-{\rm i}\bar{\nu}) \nonumber \\ (\alpha-1){\nu} & (\frac{1}{\alpha}-1)({\rm i}{\nu}) \nonumber \\ \end{array} \right)\left( \begin{array}{@{}cc@{}} u_n^- \sqrt{\gamma^-} \nonumber \\ u_l^- \sqrt{\gamma^-} \nonumber \\ \end{array} \right). \nonumber \end{align*}$$

Using (7) and (8), together with the definition of $\phi,$ we obtain the relation

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left( \begin{array}{@{}cc@{}} u_n^- \sqrt{\gamma^-} \nonumber \\ u_l^- \sqrt{\gamma^-} \nonumber \\ \end{array} \right)= \left( \begin{array}{@{}cc@{}} {\nu} & \bar{\nu} \nonumber \\ {\rm i}{\nu} & -{\rm i}\bar{\nu} \nonumber \\ \end{array} \right)\left( \begin{array}{@{}cc@{}} \phi_1^- \nonumber \\ \phi_2^- \nonumber \\ \end{array}\! \right). \nonumber \end{align*}$$

These two previous displayed equations allows us to complete the proof of the lemma.

Consider the vector $\phi$ which satisfies (4) and the following asymptotic

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{8JunA} \begin{array}{@{}l@{}} \phi_1(z,w,\lambda)={\rm e}^{\lambda (z-w)^2/4}U(z,w,\lambda)+{\rm e}^{\lambda (z-w)^2/4}o(1), \quad \nonumber \\ \phi_2(z,w,\lambda)={\rm e}^{\overline{\lambda (z-w)^2}/4}o(1), \end{array} \quad z \rightarrow \infty \nonumber \end{align} \tag{ 9 }$$

where $U(z,w,\lambda)$ is an entire function with respect to the parameter z.

We denote

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{200918A} \mu_1(z,w,\lambda)=\phi_1(z,w,\lambda) {\rm e}^{-\lambda (z-w)^2/4}, \quad \mu_2(z,w,\lambda)=\phi_2(z,w,\lambda){\rm e}^{-\overline{\lambda (z-w)^2}/4}. \nonumber \end{align} \tag{ 10 }$$

Further, we introduce some matrix functions that will establish a integral equation for $\mu$.

Due to (4), this functions fulfill the following equation on $\mathbb{C}\setminus\Gamma$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{muirbc} \left ( \begin{array}{@{}cc@{}} \bar{\partial}_z & 0 \nonumber \\ 0 & \partial_z \end{array} \right) \mu =\left( \begin{array}{@{}cc@{}} 0 & q_{12}(z){\rm e}^{-{\rm i}\,{{\rm Im}}[\lambda(z-w)^2/2]} \nonumber \\ q_{21}(z){\rm e}^{{\rm i}\,{{\rm Im}}[\lambda(z-w)^2/2]} \end{array}\right) \mu =: \tilde{q}\mu. \nonumber \end{align} \tag{ 11 }$$

On the contour $\Gamma$ they fulfill a transmission condition similar to (6), with the right-hand side being substituted by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \widetilde{A}_\lambda\mu=\frac{1}{2}\left( \begin{array}{@{}cc@{}} \alpha+\frac{1}{\alpha}-2 & (\alpha-\frac{1}{\alpha})\bar{\nu}^2 {\rm e}^{-{\rm i}\,{\rm Im}[\lambda (z-w)^2/2]} \nonumber \\ (\alpha-\frac{1}{\alpha})\nu^2{\rm e}^{{\rm i}\,{\rm Im}[\lambda (z-w)^2/2]} & \alpha+\frac{1}{\alpha}-2 \nonumber \\ \end{array} \right)\, \left(\begin{array}{@{}cc@{}} \mu_1^- \nonumber \\ \mu_2^- \end{array}\right), \nonumber \end{align*}$$

where $\mu_1^-$ and $\mu_2^-$ are the traces values of $\mu$ taken from the interior of $\Gamma$.

Through this we obtain an integral equation for $\mu$:

Proposition 2.2. Let $\mu$ be a solution of (11) given as above through a function $\phi$ which fulfills (4) and the asymptotics (9). Then $\mu$ is a solution of the following integral equation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0807A} (I+P\widetilde{A}_\lambda -D\widetilde{Q}_\lambda)\mu= \left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right), \nonumber \end{align} \tag{ 12 }$$

where $\newcommand{\e}{{\rm e}} D=\left( \begin{array}{@{}cc@{}} \bar{\partial}^{-1} & 0 \\ 0 & \partial^{-1} \\ \end{array} \right)$ with $\bar{\partial}^{-1}f(z)=\frac{1}{2\pi {\rm i}}\int_{\mathbb{C}} \frac{f(\varsigma)}{\varsigma-z}\,{\rm d}\varsigma\wedge {\rm d}\bar{\varsigma}$ and the $\partial^{-1}$ is given through the complex conjugate of the kernel $(\varsigma-z){}^{-1}$. The matrix $\widetilde{Q}_\lambda$ has the following form

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \widetilde{Q}_\lambda=\left( \begin{array}{@{}cc@{}} 0 & Q_{12} {\rm e}^{-{\rm i}\,{\rm Im}[\lambda (z-w)^2/2]} \nonumber \\ Q_{21} {\rm e}^{{\rm i}\,{\rm Im}[\lambda (z-w)^2/2]} & 0 \nonumber \\ \end{array} \right), \nonumber \end{align*}$$

where $Q_{12},\,Q_{21}$ are $L^{\infty}$ extensions of $q_{12},\,q_{21}$ to $\Gamma$. Moreover, P is a projector

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{14a} P=\left( \begin{array}{@{}cc@{}} P_+ & 0 \nonumber \\ 0& P_- \nonumber \\ \end{array} \right), \nonumber \end{align} \tag{ 13 }$$

where $P_+,\,P_-$ are the Cauchy projector and its complex adjoint, respectively:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle P_+f(w)=\frac{1}{2\pi {\rm i}} \int_{\Gamma^+} \frac{f(z)}{z-w}\,{\rm d}z, \quad P_-f(w)=\frac{1}{2\pi {\rm i}} \int_{\Gamma^+} \frac{f(z)}{\bar{z}-\bar{w}}\,{\rm d}\bar{z}, \quad w \in {\mathbb C}. \nonumber \end{align*}$$

Hereby, f  is a function defined on the contour $\Gamma$.

Proof. We use the same approach as in [22]. The following Cauchy–Green formulas hold for each $f\in C^1(\overline{\Omega})$ and an arbitrary bounded domain $\Omega$ with smooth boundary:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{2DecB} f(z)= \frac{1}{2\pi {\rm i}} \int_{\Omega} \frac{\partial f (\varsigma)}{\partial \bar{\varsigma}} \frac{1}{\varsigma-z} \,{\rm d}\varsigma\wedge {\rm d}\bar{\varsigma} + \frac{1}{2\pi {\rm i}} \int_{{\partial \Omega}} \frac{f(\varsigma)}{\varsigma - z}{\rm d}\varsigma, \quad z \in \Omega, \nonumber \end{align} \tag{ 14 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{2DecC} 0= \frac{1}{2\pi {\rm i}} \int_{\Omega} \frac{\partial f (\varsigma)}{\partial \bar{\varsigma}} \frac{1}{\varsigma-z}\,{\rm d}\varsigma\wedge {\rm d}\bar{\varsigma} + \frac{1}{2\pi {\rm i}} \int_{{\partial \Omega}} \frac{f(\varsigma)}{\varsigma - z}{\rm d}\varsigma,\quad z \not \in \overline{\Omega}. \nonumber \end{align} \tag{ 15 }$$

Denote by D_R a disk of radius R and centered at z, and take $D_R^-=D_R\setminus \overline{\mathcal{D}}$. We recall, D is the interior part of $\Gamma$. Assume that $z\in \mathcal{D}$, $f=\mu_1$ in both formulas, and $\Omega=\mathcal{D}$ in (14) and $\Omega=D_R^-$ in (15). We add the left- and right-hand sides in formulas (14) and (15). Taking the transmission condition for $\mu$ into account, we obtain for fixed w that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{11NovA} \mu_1(z,\lambda) = \frac{1}{2\pi {\rm i}} \int_{D_R\setminus\Gamma} (\tilde{q}\mu)_1(\varsigma,\lambda) \frac{1}{\varsigma-z}\,{\rm d}\varsigma\wedge {\rm d}\bar{\varsigma} + \frac{1}{2\pi {\rm i}} \int_{\Gamma^-} \frac{[\mu_1](\varsigma)}{\varsigma - z}{\rm d}\varsigma+ \frac{1}{2\pi {\rm i}} \int_{\partial D_R} \frac{\mu_1(\varsigma)}{\varsigma - z}{\rm d}\varsigma, \nonumber \end{align} \tag{ 16 }$$

where $[\mu_1]=\mu_1^- -\mu_1^+$.

Noticing that $\mu_1$ converges to U at infinity and since U is entire, then taking the limit $R\to \infty$, it follows that the last term is $U(z)$. In this way, by taking the limit, and reordering we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mu_1(z,\lambda) - \frac{1}{2\pi {\rm i}} \int_{\mathbb{C}} (\widetilde{Q}_\lambda\mu)_1(\varsigma,\lambda)\frac{1}{\varsigma-z} \,{\rm d}\varsigma\wedge {\rm d}\overline{\varsigma} +\frac{1}{2\pi {\rm i}}\int_{\Gamma^+} \frac{(\widetilde{A}_{\lambda}\mu)_1(\varsigma,\lambda)}{\varsigma-z}\,{\rm d}\varsigma = U(z). \nonumber \end{align} \tag{ 17 }$$

This equation together with similar computations for $z \in D_R^{-}$ and showing the case for $\mu_2$ (similarly by taking the adjoint Cauchy–Green formulas) we obtain the desired integral equation.

Let $1<p<\infty$, R  >  0 and $\newcommand{\e}{{\rm e}} f = \left(\begin{array}{@{}c@{}} f_1 \\ f_2 \end{array}\right)$ be a vector function.

To define our spaces we keep in mind the notation introduced in [23]. Denote by $L^{\infty}_z(B)$ the space of bounded functions of $z\in \mathbb{C}$ with values in a Banach space B. Thus, picking $B=L^{\,p}_{\lambda}(|\lambda|>R)$ we introduce the first space

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle {\mathcal H}_1^{\,p}:=\left\{\,f: f_1,\,f_2\;\, {\rm continuous ~ functions in}\, L^{\infty}_z\left(L^{\,p}_{\lambda}(|\lambda|>R)\right) \cap L^{\infty}_z\left(L^{\infty}_{\lambda}(|\lambda|>R)\right)\right\}.\nonumber \end{align*}$$

To simplify the notation ahead, we introduce the following function space:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle S=\left\{g:\Gamma\times\{\lambda\in\mathbb{C}:|\lambda|>R\}\rightarrow \mathbb{C}^2 \;\; {{\rm s.t.}}\; \sum_{i\in\{1,2\}} \int_{|\lambda|>R} \int_{\Gamma} |g_i(z,\lambda)|^{\,p}\,{\rm d}|z| {\rm d}\sigma_\lambda < \infty\right\},\nonumber \end{align*}$$

where ${\rm d}\sigma_{\lambda}$ is the Lebesgue measure in $\mathbb{R}^2$ (similarly we define ${\rm d}\sigma_{z}$).

Following the idea of Hardy spaces and to obtain desirable properties at the contour $\Gamma$ we define the second space through the projector P in (13) by:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle {\mathcal H}_2^{\,p}:= \left\{F\in \mathcal{R}(P): \sum_{i\in\{1,2\}} \int_{|\lambda|>R}\int_{\Gamma} \left|F^-_i(z,\lambda)\right|^{\,p}\,{\rm d}|z|{\rm d}\sigma_{\lambda}<\infty\right\} \nonumber \end{align*}$$

where by $\mathcal{R}(P)$ we mean the range of the matrix projector P with domain S. Hence we have that for $F \in \mathcal{H}^{\,p}_2$ there exists a function $f\in S$ such that F  =  Pf and in $\Gamma$ it fulfills F⁻  =  f . Moreover, this allows to consider this space with the norm

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|F\|_{\mathcal{H}^{\,p}_2}^{\,p}:=\sum_{i\in\{1,2\}} \int_{|\lambda|>R} \int_{\Gamma} |F^-_i(z,\lambda)|^{\,p}\,{\rm d}|z| {\rm d}\sigma_\lambda= \sum_{i\in\{1,2\}} \int_{|\lambda|>R} \int_{\Gamma} |f_i(z,\lambda)|^{\,p}\,{\rm d}|z| {\rm d}\sigma_\lambda.\nonumber \end{align*}$$

Finally, the space we are going to work with is given as ${\mathcal H}^{\,p} = {\mathcal H}^{\,p}_1 + {\mathcal H}^{\,p}_2$ endowed with the norm

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{2107B} \|t\|_{{\mathcal H}^{\,p}}=\inf_{{u+v=t \atop u \in {\mathcal H}^{\,p}_1, v \in {\mathcal H}^{\,p}_2}} \max (\|u\|_{{\mathcal H}^{\,p}_1}, \|v\|_{{\mathcal H}^{\,p}_2 }). \nonumber \end{align} \tag{ 18 }$$

Let us remind that the operations of intersection and union of two Banach spaces are correctly defined if all terms can be continuously embedded into a common locally convex space. In our situation this common locally convex space will be a space endowed with the semi-norms

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_{|\lambda|>R} \int_{{\mathcal O}} \frac{1}{|\lambda|^2}|f(z,\lambda)|\,{\rm d}\sigma_z {\rm d}\sigma_\lambda. \nonumber \end{align*}$$

If $f \in {\mathcal H}_1^{\,p}$ the embedding is evident. For $f \in {\mathcal H}_2^{\,p}$ we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|Pf\|_{L^{\,p}({\mathcal O})} \leqslant C \|f\|_{L^{\,p}(\Gamma)}, \nonumber \end{align*}$$

so that $\left[\|Pf\|_{L^{\,p}({\mathcal O})} \right]^{\,p} \leqslant \left[\|f\|_{L^{\,p}(\Gamma)}\right]^{\,p}$ and

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int \left ( \int_{{\mathcal O}} |Pf(z)|^{\,p} \,{\rm d}\sigma_z \right) \,{\rm d}\sigma_\lambda \leqslant \int [\|f\|_{L^{\,p}(\Gamma)}]^{\,p}\, {\rm d}\sigma_\lambda = \int_{|\lambda|>R} \int_{\Gamma} |f(z,\lambda)|^{\,p} \,{\rm d}|z|{\rm d}\sigma_\lambda. \nonumber \end{align*}$$

The boundedness of each semi-norm follows from the continuity of the embedding of $L^{\,p}({\mathcal O})$ into $L^1({\mathcal O})$.

Lemma 3.1. The operators $\widehat{P}_\pm: f \rightarrow (Pf)|_{\Gamma^\pm}$ are bounded in the space with norm

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left[\int_{|\lambda|>R} \int_{\Gamma} |f(z,\lambda)|^{\,p}\,{\rm d}z {\rm d}\sigma_\lambda\right]^{1/p}.\nonumber \end{align*}$$

Proof. During the proof the sign $\pm$ in the projectors will be omitted. From the continuity of Cauchy projectors in $L^{\,p}(\Gamma)$ follows

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|\widehat{P} f \|_{L^{\,p}(\Gamma)} \leqslant C \|f \|_{L^{\,p}(\Gamma)}. \nonumber \end{align*}$$

and therefore

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left (\|\widehat{P} f \|_{L^{\,p}(\Gamma)} \right)^{\,p}\leqslant C^{\,p} \left ( \|f \|_{L^{\,p}(\Gamma)} \right)^{\,p}. \nonumber \end{align*}$$

Finally

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|P\widehat{P}f\|_{{\mathcal H}^{\,p}_2}= \int_{|\lambda|>R} \left (\|\widehat{P} f \|_{L^{\,p}(\Gamma)} \right)^{\,p} \,{\rm d}\sigma_\lambda \leqslant C^{\,p} \int_{|\lambda|>R} \left( \|f \|_{L^{\,p}(\Gamma)} \right)^{\,p} \,{\rm d}\sigma_\lambda = C^{\,p} \int_{|\lambda|>R} \int_{\Gamma} |f(z,\lambda)|^{\,p} \,{\rm d}|z|{\rm d}\sigma_\lambda .\nonumber \end{align*}$$

Lemma 3.2. Let $u \in {\mathcal H}^{\,p}_1$. Then $P(u|_{\Gamma}) \in {\mathcal H}^{\,p}_2$.

Proof. From the definition of ${\mathcal H}^{\,p}_1$, combined with the fact that u is a continuous function, we get

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|u\|_{L^{\,p}_\lambda} \in L^\infty_z(\Gamma). \nonumber \end{align*}$$

Since $\Gamma$ is a bounded set, the L^p norm does not exceed (up to a constant) the $L^\infty$ norm and, therefore

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|\|u\|_{L^{\,p}_\lambda}\|_{L^{\,p}_z(\Gamma)} \leqslant C \|u\|_{{\mathcal H}^{\,p}_1}. \nonumber \end{align*}$$

Now we just note the left-hand side of the above inequality is the norm ${\mathcal H}^{\,p}_2$ norm.

Multiplying equation (12) by $I+D\widetilde{Q}_\lambda$ we get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0907H} (I+M)\mu= (I+D\widetilde{Q}_\lambda) \left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right) \nonumber \end{align} \tag{ 19 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0907I} M=P\widetilde{A}_\lambda +D\widetilde{Q}_\lambda P\widetilde{A}_\lambda -D\widetilde{Q}_\lambda D\widetilde{Q}_\lambda. \nonumber \end{align} \tag{ 20 }$$

Lemma 3.3. Let $\,{\rm Re}(\gamma)\geqslant c >0$ and $\widetilde{A}_{\lambda}$ in $L^{\infty}(\Gamma)$. Then the operators $D\widetilde{Q}_\lambda P\widetilde{A}_\lambda ,\,D\widetilde{Q}_\lambda D\widetilde{Q}_\lambda$ are bounded in ${\mathcal H}^{\,p},\, p>1$.

Moreover, if R  >  0 is large enough they are contractions and if $\alpha-1$ is small enough in $L^{\infty}(\Gamma)$ then $P\widetilde{A}_{\lambda}$ is a contraction in $\mathcal{H}^{\,p}, \,p>1$.

Proof. In order to estimate $\|(D\widetilde{Q}_\lambda P\widetilde{A}_\lambda)t\|_{{\mathcal H}^{\,p}}$ and $\|(D\widetilde{Q}_\lambda D\widetilde{Q}_\lambda)t\|_{{\mathcal H}^{\,p}}$ (recall definition 18) we consider the representation $t=u+v$ where the infimum is (almost) achieved. It is easy to see that the desired estimate follows from the fact that these operators are a contraction in each of the spaces, ${\mathcal H}_1^{\,p}$ and ${\mathcal H}_2^{\,p}$. This fact can be shown as follows.

In lemma 2.1 of [23] it was proved that the operator $D\widetilde{Q}_\lambda D\widetilde{Q}_\lambda$ is bounded in ${\mathcal H}^{\,p}_1$. The proof that it is also a contraction in $\mathcal{H}^{\,p}_2$ and the statement for $D\widetilde{Q}_\lambda P\widetilde{A}_\lambda$ follows in a similar manner.

Hereby, we show the case for $D\widetilde{Q}_\lambda P \widetilde{A}_{\lambda}$. By definition we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle D\widetilde{Q}_\lambda P\widetilde{A}_{\lambda} u(z)=\left\{\begin{array}{@{}ccc@{}} \int_{\Gamma} [\widetilde{A}_{\lambda}u]_2(z_2)G_1(z,z_2,\lambda,w)\,{\rm d}z_2 \nonumber \\ \int_{\Gamma} [\widetilde{A}_{\lambda}u]_1(z_2)G_2(z,z_2,\lambda,w)\,{\rm d}z_2 \end{array}\right.\nonumber \end{align*}$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{AAA} G(z,z_2,\lambda,w)=\left(\begin{array}{@{}c@{}} G_1 \nonumber \\ G_2 \end{array}\right) =\left\{\begin{array}{@{}cc@{}} (2\pi {\rm i})^{-2}\int_{{\mathcal O}} \frac{{\rm e}^{-{\rm i}\,{\rm Im}[\lambda(z_1-w)^2]/2}}{{z_1}-{z}} \frac{{Q}_{12}(z_1)}{\bar{z}_2-\bar{z}_1} \, {\rm d}{\sigma_{z_1}} \nonumber \\ (2\pi {\rm i})^{-2}\int_{{\mathcal O}} \frac{{\rm e}^{{\rm i}\,{\rm Im}[\lambda(z_1-w)^2]/2}}{{\bar{z}_1}-\bar{z}} \frac{{Q}_{21}(z_1)}{{z}_2-{z}_1} \, {\rm d}{\sigma_{z_1}} \end{array}\right. . \nonumber \end{align} \tag{ 21 }$$

By following a similar estimation on the proof of lemma 2.1 of [23] we obtain by the stationary phase approximation:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sup_{{z\atop |\lambda|>R}} \|G_i(z,\cdot,\lambda,w)\|_{L^q_{z_2}(\Gamma)} \leqslant \frac{1}{R}, ~ \quad 1/p+1/q=1 ~ {{\rm and}} ~i=1,2. \nonumber \end{align*}$$

Thus

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle |D\widetilde{Q}_\lambda P \widetilde{A}_{\lambda}u|(z)\leqslant \|G(z,\cdot,\lambda,w)\|_{L^q_{z_2}(\Gamma)} \|\widetilde{A}_{\lambda}u\|_{L^{\,p}_{z_2}(\Gamma)}. \nonumber \end{align*}$$

Then, we have for

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|D\widetilde{Q}_\lambda P\widetilde{A}_{\lambda} u(z)\|_{L^{\,p}_\lambda} \leqslant \|G(z,\cdot,\lambda,w)\|_{L^q_{z_2}(\Gamma)} \|\widetilde{A}_{\lambda}\|_{L^\infty(\Gamma)} \|u\|_{{\mathcal H}^{\,p}_2} \nonumber \end{align*}$$

where we used the fact that $u \in {\mathcal H}^{\,p}_2$ is the same as $\|u\|_{L^{\,p}_{z_2}(\Gamma)} \in L^{\,p}_\lambda$. The final estimate follows from the definitions of both spaces and the above uniform bound on G_i.

If we take R  >  0 large enough then it follows that $D\widetilde{Q}_{\lambda}D\widetilde{Q}_{\lambda}$ and $D\widetilde{Q}_{\lambda}P\widetilde{A}_{\lambda}$ are contractions in $\mathcal{H}^{\,p}$ as long as $\|\widetilde{A}\|_{L^{\infty}(\Gamma)}$ is finite.

By the definition of $\mathcal{H}^{\,p}$ the boundedness of $P\widetilde{A}_{\lambda}$ follows from the usual L^p boundedness. Since this operator will not have the same dependence on $\lambda$ as the others we need the jump to be close enough to 1 so that the supremum norm in z of $\widetilde{A}_{\lambda}$ on $\Gamma$ is small enough and possibilitates the norm of the whole operator to be less than 1.

A rough estimate for this norm is given in terms of the jump by:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \|\widetilde{A}_{\lambda}\|_{L^{\infty}(\Gamma)}\leqslant 2\left|\alpha-1\right|\left(1+\left|\frac{1}{\alpha}\right|\right)\leqslant 4\epsilon,\nonumber \end{align*}$$

where $\newcommand{\e}{{\rm e}} \epsilon>0$ is an upper bound for $\left|\alpha-1\right|$. Hence for $P\widetilde{A}_{\lambda}$ to be a contraction on $\mathcal{H}^{\,p}$ we need that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle |\alpha-1| \leqslant \frac{1}{4\|P\|_{\mathcal{H}^{\,p}}}.\nonumber \end{align*}$$

Let $w \in {\mathcal O}$ be a fixed point. For the asymptotics (9) we can take any entire function. In our approach, we take this entire functions to be:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0907A} U(z,w,\lambda)={\rm e}^{\ln |\lambda| \lambda_{{\mathcal O}}(z-w)^2}, \nonumber \end{align} \tag{ 22 }$$

where $z\in \mathbb{C}$ and $\lambda_{{\mathcal O}}$ is a parameter.

These functions lead us to the concept of admissible points.

We recall here their definition: we say that a point $w \in \mathcal{O}$ is an admissible point, if there is a number $\lambda_{{\mathcal O}}\in\mathbb{C}$ such that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle A&:=\sup_{z \in \overline{{\mathcal O}}} {\rm Re}[ \lambda_{{\mathcal O}}(z-w)^2]< 1/2,\nonumber \\ B&:= \sup_{z \in \overline{{\mathcal D}}} {\rm Re}[ \lambda_{{\mathcal O}}(z-w)^2] < -1/2. \nonumber \end{align*}$$

Moreover, if w is an admissible point and A and B fulfills $\newcommand{\e}{{\rm e}} A=1/2-\epsilon_1, \, B=-1/2-\epsilon_2,$ with $\newcommand{\e}{{\rm e}} \epsilon_2-\epsilon_1>0$, we further say that w is a proper admissible point.

Note. The set of admissible points is not empty. In order to see this we consider a boundary point $w_0 \in \partial {\mathcal O}$ which belongs also to the convex hull of ${\mathcal O}$. It is easy to see that all interior points $w \in {\mathcal O}$ near the w₀ would be admissible.

We will not try to give a general geometric description of admissible points. Instead, we are only aiming to show the viability of the concept.

Denote

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{1107A} f=\mu - (I+D\widetilde{Q}_\lambda)\left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right), \nonumber \end{align} \tag{ 23 }$$

where $\mu$ is defined in (10).

The vector f  satisfies the equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{1007P} (I+M)\,f= -M(I+D\widetilde{Q}_\lambda) \left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right). \nonumber \end{align} \tag{ 24 }$$

We know already that for R  >  0 large enough the operator in the left-hand side of this equation is a contraction in ${\mathcal H}^{\,p},\, p>1$ and below we show that in fact we have for the right-hand side:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{1007Q} \frac{1}{|\lambda|^A} M(I+D\widetilde{Q}_\lambda) \left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right) \in {\mathcal H}^{\,p}, \quad p>2. \nonumber \end{align} \tag{ 25 }$$

Therefore, we get the following statement

Lemma 3.4. For any p   >  2 and R large enough such that U is given in terms of a proper admissible point w we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0907F} \frac{1}{|\lambda|^A} \left [ \mu - (I+D\widetilde{Q}_\lambda)\left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right) \right ] \in {\mathcal H}^{\,p}. \nonumber \end{align} \tag{ 26 }$$

Proof. We start by showing that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\U}{\left(\!\begin{array}{c}U \nonumber \\0\end{array}\!\right)} \displaystyle \label{Hp}\frac{1}{|\lambda|^A}\U \in \mathcal{H}_2^{\,p} \nonumber \end{align} \tag{ 27 }$$

and

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\U}{\left(\!\begin{array}{c}U \nonumber \\0\end{array}\!\right)} \displaystyle \frac{1}{|\lambda|^A}MD\tilde{Q}_{\lambda}\U \in \mathcal{H}_1^{\,p}.\nonumber \end{align*}$$

Since M is a contraction for R  >  0 big enough we are going to obtain (25) and the result will immediately follows for p   >  2.

To show (27) we refer to the following simple estimate

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \Bigg[\int_{|\lambda| > R}\int_{\Gamma}\Bigg|\frac{1}{|\lambda|^A}{\rm e}^{\ln |\lambda|\lambda_s(z-w)^2}\Bigg|\,{\rm d}|z|\,{\rm d}\sigma_{\lambda}\Bigg]^{1/p} &\leqslant \Bigg[\int_{|\lambda| > R}\int_{\Gamma}\Bigg|\frac{1}{|\lambda|^A} |\lambda|^B\Bigg|^{\,p}\,{\rm d}|z|\,{\rm d}\sigma_{\lambda}\Bigg]^{1/p} \nonumber \\ & = \Bigg[\int_{|\lambda| > R}\int_{\Gamma}\Bigg|\frac{1}{|\lambda|^{1+(\epsilon_2-\epsilon_1)}} \Bigg|^{\,p}\,{\rm d}|z|\,{\rm d}\sigma_{\lambda}\Bigg]^{1/p} < \infty \nonumber \end{align*}$$

For the second statement we need to dismantle M into its various parts and show that the statement holds for each one of them. The trick is always the same, so we will only show one of the computations, namely the one corresponding to the term $\newcommand{\e}{{\rm e}} \newcommand{\U}{\left(\!\begin{array}{c}U \\0\end{array}\!\right)} \frac{1}{|\lambda|^A}(D\tilde{Q}_{\lambda}){}^3\U$. By lemma 3.6 we get

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &\sup_{z \in {\mathcal O}} \Bigg[\int_{|\lambda|>R}\Bigg|\frac{1}{|\lambda|^A}\int_{\mathcal{O}}\frac{{\rm e}^{{\rm i}\,{\rm Im}(\lambda(z_1-w)^2)}}{\overline{z_1-z}}Q_{21}(z_1)\int_{\mathcal{O}}\frac{{\rm e}^{-{\rm i}\,{\rm Im}(\lambda(z_2-w)^2)}}{z_2-z_1}Q_{12}(z_2) \cdot \nonumber \\ & \hspace{7cm}\cdot\int_{\mathcal{O}}\frac{{\rm e}^{\rho(z_3)}}{z_3-z_2}Q_{21}(z_3)\,{\rm d}\sigma_{z_3}\,{\rm d}\sigma_{z_2}\,{\rm d}\sigma_{z_1}\Bigg|^{\,p}\,{\rm d}\sigma_{\lambda}\Bigg]^{1/p} \nonumber \\ &\leqslant \sup_{z \in {\mathcal O}} \Bigg[C ||Q||_{L^{\infty}}^3\int_{\mathcal{O}} \int_{\mathcal{O}}\frac{1}{|z_1-z|}\frac{1}{|z_2-z_1|}\frac{1}{|z_2-w|^{1-\delta}} \, {\rm d}\sigma_{z_2}\,{\rm d}\sigma_{z_1}\Bigg] < C' \nonumber \end{align*}$$

Thus, the result (25) follows, and in consequence also (26) holds from (23) and (24).

Let $w\in\mathcal{O}$ be an admissible point. We consider the function

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0907C} {\rm e}^{\overline{\ln |\lambda| \lambda_s(z-w)^2}}, \nonumber \end{align} \tag{ 28 }$$

where the number $\lambda_s$ is chosen as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0907D} \sup_{z \in \overline{{\mathcal O}}} {\rm Re}[ \lambda_s(z-w)^2]<1/2, \quad \sup_{z \in \overline{{\mathcal D}}} {\rm Re}[ \lambda_s(z-w)^2]<-1/2. \nonumber \end{align} \tag{ 29 }$$

A point w can be admissible to more than one spectral parameter. To define the scattering data we want to use the above exponentials depending on $\lambda_s$. Since $\mu$ fulfills an asymptotic with the spectral parameter being $\lambda_{{\mathcal O}}$ we also define our scattering data with respect to this spectral parameter, i.e. $\lambda_s=\lambda_{{\mathcal O}}$. However, this is not a requirement and $\lambda_s$ could have been one of the other parameters which makes w admissible. We just fix it like this to simplify the proofs ahead.

Consider now our scattering data

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{1007S} h(\lambda,w)= \int_{\partial {\mathcal O}} \overline{{\rm e}^{\ln |\lambda| \lambda_s(z-w)^2}} \mu_2(z){\rm d}\bar{z}. \nonumber \end{align} \tag{ 30 }$$

Using Green's theorem

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_{\partial {\mathcal O}} f \, {\rm d}\bar{z} = -2{\rm i} \int_{{\mathcal O}} \partial {f} \, {\rm d}{\sigma_{z}} \nonumber \end{align*}$$

we can see that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0907E} h(\lambda,w)= \int_{\Gamma^+} \overline{{\rm e}^{\ln |\lambda| \lambda_s(z-w)^2}} \mu_2(z){\rm d}\bar{z} + \int_{{\mathcal O} \setminus\overline{\mathcal{D}}} \overline{{\rm e}^{\ln |\lambda| \lambda_s(z-w)^2}} {\rm e}^{-{\rm i}\,{\rm Im} [\lambda (z-w)^2]/2} q_{21}(z)\mu_1(z) {\rm d}\sigma_z. \nonumber \end{align} \tag{ 31 }$$

This formula gives raise to an operator that we denote by T and it is defined by

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle T[G](\lambda)= \int_{{\mathcal O} \setminus\overline{\mathcal{D}}} \overline{{\rm e}^{\ln |\lambda| \lambda_s(z-w)^2}} {\rm e}^{-{\rm i}\,{\rm Im} [\lambda (z-w)^2]/2} q_{21}(z)G(z) {\rm d}\sigma_z. \nonumber \end{align*}$$

From our representation for the solution $\mu$ (23) and the fact that the matrix $\widetilde{Q}_{\lambda}$ is off-diagonal we get

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle T\left [ \left ( (I+D\widetilde{Q}_\lambda)\left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right) \right)_1\right ] = T[U]. \nonumber \end{align*}$$

This allows us to state our main theorem.

Theorem 3.5. Let the potential q being given through (5) for an admittivity $\gamma$ satisfying ${\rm Re}(\gamma)\geqslant c>0$ and $\gamma\in W^{2,\infty}(\mathcal{D})\cap W^{2,\infty}(\mathcal{O}\setminus\overline{\mathcal{D}})$. If the jump $\alpha-1$ is small enough in $L^{\infty}(\Gamma)$, and w is a proper admissible point for a spectral parameter $\lambda_s$, then

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{210918A} \frac{\lambda_s}{4\pi^2\ln 2}\lim_{R\to\infty}\int_{R<|\lambda|<2R} |\lambda|^{-1} \, h(\lambda,w)\, {\rm d}\sigma_\lambda=q_{21}(w). \nonumber \end{align} \tag{ 32 }$$

The proof of this theorem requires some additional results concerning the behavior of $h(\lambda,w)/|\lambda|.$ These results will be given in the form of three lemmas which we establish in the next section.

We start by presenting a result which we need afterwards. For its proof we refer to appendix A. Consider two arbitrary numbers $\lambda_0,\,w \in {\mathbb C}$, denote $\rho(z)= -{\rm i}\,{\rm Im}[\lambda(z-w){}^2]/2\,+$ $\ln |\lambda| \lambda_0(z-w){}^2$, and let $A_0=\sup_{z \in {\mathcal O}}\, {\rm Re}[ \lambda_0(z-w){}^2].$

Lemma 3.6. Let $z_1 \in {\mathbb C}\setminus\{w\}$, $p > 2,$ and $\varphi \in L^\infty$ with compact support. Then

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left \|\frac{1}{|\lambda|^{A_0}}\int_{{\mathbb C}} \varphi(z)\frac{{\rm e}^{\rho(z)} }{z-z_1} \, {\rm d}{\sigma_{z}} \right \|_{L^{\,p}_\lambda({\mathbb C})} \leqslant C\frac{\|\varphi\|_{L^\infty}}{|z_1-w|^{1-\delta}}, \nonumber \end{align*}$$

where the constant C depends only on the support of $\varphi$ and on $\delta =\delta(\,p)>0$.

To study the main term in (32), we have the following lemma.

Lemma 3.7. Let $\varphi \in W^{1,\infty}(\Omega)$, and ${\rm supp}(\varphi) \subset \Omega$, where $\Omega$ is a domain in ${\mathbb R}^2$ and $w \in \Omega$ such that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0807C} \sup_{z \in \overline{\Omega}} {\rm Re} (z-w)^2 <1. \nonumber \end{align} \tag{ 33 }$$

Then the following asymptotic holds

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0807B} \int_{\Omega} {\rm e}^{-{\rm i}\,{\rm Im} (\lambda (z-w)^2)+\ln|\lambda|(z-w)^2} \varphi(z) {\rm d}\sigma_z = \frac{2\pi}{|\lambda|}\varphi(w)+R_w(\lambda), \nonumber \end{align} \tag{ 34 }$$

where $|\lambda|^{-1}R_w \in L^1(\lambda :|\lambda|>R)$.

Proof. Consider two domains

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle I_1=\{z\in\Omega ~ : ~ |z-w| < 3\varepsilon\} \quad{{\rm and}} \quad I_2=\{z\in\Omega ~ : ~ |z-w| > \varepsilon \}, \nonumber \end{align*}$$

where $\varepsilon>0$ is an a priori chosen arbitrarily small but fixed number. Furthermore, we pick two functions $\delta_1$ and $\delta_2$ with supports I₁ and I₂, respectively, such that $\newcommand{\e}{{\rm e}} \delta_1+\delta_2 \equiv 1$ in ${\mathcal O}$. Moreover, we assume that $\delta_1(z-w)$ is represented as a product of $\widehat{\delta}_1(x)\widehat{\delta}_1(y)$ and that the function $\widehat{\delta}_1(x)$ decreases monotonically as $|x|$ grows.

The integrand is multiplied by $(\delta_1+\delta_2)$ and this naturally splits the integral into two terms. The term corresponding to $\delta_2$ can be integrated by parts once and then the required estimate follows from the Hausdorff–Young inequality (B.1) for p   =  q  =  2. We also use here the fact that the inequality (33) is strict.

Now, we consider the term corresponding to integration against $\delta_1$. This term will be divided into two parts as well correspondent to the representation

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \delta_1(z) \varphi(z)=\delta_1(z)\varphi(w)+ \delta_1(z)(\varphi(z)-\varphi(w)). \nonumber \end{align*}$$

Keeping in mind the properties of $\delta_1$ and the fact that $\varphi \in W^{1,\infty}$ is a Lipschitz function the second part can be treated as in lemma 3.6, i.e. we make a change of variables u  =  (z  −  w)² and thus ${\rm d}\sigma_{z}=\frac{1}{4|u|}{\rm d}\sigma_{u}$. From this we obtain two integrals due to the splitting of the domain. In each one of them the change of variables generates a singularity of total order $|u|$. In both integrals we can apply integration by parts. We will obtain an area integral and a contour integral. On the area integral we have a singularity of total order |u|^3/2. Hence, we can apply Hausdorff–Young inequality for the Laplace transform for p   =  4/3 and obtain here the required estimate. For the contour integral we can apply the one-dimensional Hausdorff-inequality to the Laplace transform and obtain the needed estimate.

To the first part we consider the change of variables $y={\sqrt{|\lambda|}}(z-w)$. Due to the separation of variables in $\delta_1$ the asymptotic of

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{0807d} \frac{\varphi(w)}{|\lambda|} \int {\rm e}^{-{\rm i}\,{\rm Im} y^2+\frac{\ln|\lambda|}{|\lambda|}y^2} \delta_1\left(\left|w+\frac{y}{\sqrt{|\lambda|}}\right|\right) {\rm d}\sigma_y \nonumber \end{align} \tag{ 35 }$$

follows from the formula

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{2207A} \int_0^{\lambda^{1/2}\delta} {\rm e}^{-{\rm i}x^2+\frac{\ln|\lambda|}{|\lambda|}x^2} \widehat{\delta}_1\left(\frac{|x|}{\sqrt{|\lambda|}}\right) {\rm d}x = \frac{1}{\sqrt{2\pi}}(1+o(1)), \quad \lambda \rightarrow \infty. \nonumber \end{align} \tag{ 36 }$$

This can be proven in the following way: consider the change of variables

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle x^2=t, \;\,g(t):=\widehat{\delta}_1\left(\frac{|x(t)|}{\sqrt{|\lambda|}}\right)\nonumber \end{align*}$$

then, we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_0^{\lambda^{1}\delta^2} {\rm e}^{-{\rm i}t+\frac{\ln|\lambda|}{|\lambda|}t} \frac{1}{\sqrt t}g(t) {\rm d}t \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle =\int_0^1 {\rm e}^{-{\rm i}t+\frac{\ln|\lambda|}{|\lambda|}t} \frac{1}{\sqrt t}g(t) {\rm d}t + \frac{1}{-{\rm i}+\frac{\ln|\lambda|}{|\lambda|}}\int_1^{\lambda \delta^2} \left( {\rm e}^{-{\rm i}t+\frac{\ln|\lambda|}{|\lambda|}t} \right)' \frac{1}{\sqrt t}g(t) {\rm d}t. \nonumber \end{align*}$$

For the second term we obtain

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \frac{1}{-{\rm i}+\frac{\ln|\lambda|}{|\lambda|}}\int_1^{\lambda \delta^2} \left( {\rm e}^{-{\rm i}t+\frac{\ln|\lambda|}{|\lambda|}t} \right)' \frac{1}{\sqrt t}g(t) {\rm d}t = \left . \frac{1}{-{\rm i}+\frac{\ln|\lambda|}{|\lambda|}}\left( {\rm e}^{-{\rm i}t+\frac{\ln|\lambda|}{|\lambda|}t} \right) \frac{g(t)}{\sqrt t} \right |_1^{\lambda\delta^2} \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle +\frac{1}{-{\rm i}+\frac{\ln|\lambda|}{|\lambda|}}\int_1^{\lambda \delta^2} {\rm e}^{-{\rm i}t+\frac{\ln|\lambda|}{|\lambda|}t} \frac{1}{2t^{3/2}}g(t) {\rm d}t \nonumber \end{align*}$$

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle =\frac{-1}{\rm -i}\left( {\rm e}^{-{\rm i}} \right) \frac{g(1)}{\sqrt 1} +\frac{1}{\rm -i}\int_1^{\lambda \delta^2} {\rm e}^{-{\rm i}t} \left ( \frac{g(t)}{2t^{1/2}} \right)' {\rm d}t +o(1), \quad \lambda \to \infty. \nonumber \end{align*}$$

We used here the fact that the last integral is absolutely convergent (g has a finite support) and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{2207D} \sup_{z \in I_1} \left |{\rm e}^{\frac{\ln|\lambda|}{|\lambda|}y^2} -1 \right | = o(1), \quad \lambda \rightarrow \infty. \nonumber \end{align} \tag{ 37 }$$

Therefore, we get

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_0^{\lambda \delta^2} {\rm e}^{-{\rm i}x^2+\frac{\ln|\lambda|}{|\lambda|}x^2} f(x) {\rm d}x = \int_0^{\lambda \delta^2} {\rm e}^{-{\rm i}t} \frac{1}{\sqrt t}g(t) {\rm d}t = \int_0^{\infty} {\rm e}^{-{\rm i}t} \frac{1}{\sqrt t}g(t) {\rm d}t +o(1). \nonumber \end{align*}$$

Now the result of our lemma is an immediate consequence of this formula.

To prove our asymptotic formula of the scattering data, we will substitute $\mu$ with the help of (23). This will leave some terms which need to vanish as $|\lambda|\rightarrow +\infty$ in order to obtain the desired formula through lemma 3.7. In this sense, the two lemmas that follow assure this remaining terms are integrable in $\lambda$ and, therefore, their impact vanishes as we take the limit.

Lemma 3.8. For some p   <  2, with R large enough and f  defined as in (23), we get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{1007A} \frac{1}{|\lambda|}T\left [ M(I+D\widetilde{Q}_\lambda)\left( \begin{array}{@{}c@{}} U \nonumber \\ 0 \nonumber \\ \end{array} \right) \right ] \in L^{\,p}(\lambda : |\lambda|>R), \quad \nonumber \end{align} \tag{ 38 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{1107B} \frac{1}{|\lambda|}T[Mf] \in L^{\,p}(\lambda : |\lambda|>R). \nonumber \end{align} \tag{ 39 }$$

Proof. Given the structure of $M=P\tilde{A}_{\lambda}+D\tilde{Q}_{\lambda}-D\tilde{Q}_{\lambda}D\tilde{Q}_{\lambda}$ and that $\frac{1}{|\lambda|}T$ is a linear operator, it is enough to show that each term applied to both, $\newcommand{\e}{{\rm e}} \newcommand{\U}{\left(\!\begin{array}{c}U \\0\end{array}\!\right)} \U$ and $\newcommand{\e}{{\rm e}} \newcommand{\U}{\left(\!\begin{array}{c}U \\0\end{array}\!\right)} D\tilde{Q}_{\lambda}\U$, belongs to $L^{\,p}(\lambda:|\lambda|> R)$.

We look directly at the computations of each term. By using Fubini's theorem, Minkowski integral inequality, Hölder inequality, and lemma 3.6 we can show that all of these terms are in fact in $L^{\,p}(\lambda:|\lambda|> R)$. Since the computations for each term follow roughly the same lines, and for the convenience of the reader, we present just the computation in one of these cases, the computations of the remaining terms being analogous, with special attention to the convergence of the integrals.

We look at the term

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\U}{\left(\!\begin{array}{c}U \nonumber \\0\end{array}\!\right)} \displaystyle \frac{1}{|\lambda|}T\Bigg[D\tilde{Q}_{\lambda}D\tilde{Q}_{\lambda}\U\Bigg] \in L^{\,p}(\lambda:|\lambda|> R).\nonumber \end{align*}$$

Let us denote $\rho(z)={\rm i}\,{\rm Im}[\lambda(z-w){}^2]/2+\ln|\lambda|\lambda_s(z-w){}^2$ and $A=S=\sup_{z \in \overline{\mathcal{O}}} \,$ $[\lambda(z-w){}^2] < 1/2$.

With these calculations we obtain (38). To show (39) we have that $\frac{1}{|\lambda|^A}f \in \mathcal{H}^{\,p}$, for p   >  2, by lemma 3.4. We consider T applied to each term of M. Again, we present only the computations for the case $\frac{1}{|\lambda|}T[D\tilde{Q}_{\lambda}D\tilde{Q}_{\lambda}f]$, since the other computations are analogous, with special attention to the behavior of $\frac{1}{|\lambda|^A}f$. In the same spirit, we only present the calculation for the first term of the vector.

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle &\Bigg[\int_{|\lambda| > R} \Bigg|\frac{1}{|\lambda|}\int_{\mathcal{O}\setminus\overline{\mathcal{D}}} {\rm e}^{\overline{\rho(z)}}Q_{21}(z) \int_{\mathcal{O}} \frac{{\rm e}^{-{\rm i}\,{\rm Im}(\lambda(z_1-w)^2)}}{z_1-z}Q_{12}(z_1) \nonumber \\ & \hspace{2cm}\cdot \int_{\mathcal{O}}\frac{{\rm e}^{{\rm i}\,{\rm Im}(\lambda(z-w)^2)}}{\overline{z_2-z_1}}Q_{21}(z_2)\,f_1(z_2)\,{\rm d}\sigma_{z_2}\,{\rm d}\sigma_{z_1}\,{\rm d}\sigma_{z}\Bigg|^{\,p}\,{\rm d}\sigma_{\lambda}\Bigg]^{1/p} \nonumber \\ & \leqslant C\|Q \|_{L^{\infty}}^3\int_{\mathcal{O}}\int_{\mathcal{O}} \frac{1}{|z_2-z_1|} \frac{1}{|z_1-w|^{1-\delta}} \left\|\frac{1}{|\lambda|^{A}}f_1(z_2)\right\|_{L^{2p}_{\lambda}}\,{\rm d}\sigma_{z_2}\,{\rm d}\sigma_{z_1} < \infty \nonumber \end{align*}$$

The boundedness of the last integral follows from the fact that $\frac{1}{|\lambda|^A}f \in \mathcal{H}^{\,p}$ implies its boundedness with respect to the z variable.

Lemma 3.9. For R large enough, and w being a proper admissible point, we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \frac{1}{|\lambda|} \int_{\Gamma^+} {\rm e}^{\overline{\ln |\lambda| \lambda_s(z-w)^2}} \mu_2(z){\rm d}\bar{z} \in L^1(|\lambda|>R). \nonumber \end{align*}$$

Proof. We divide the integral

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{121A} \frac{1}{|\lambda|}\int_{\Gamma^{+}} {\rm e}^{\overline{\ln |\lambda|\lambda_s(z-w)^2}}\mu_2(z) {\rm d}\bar{z}, \nonumber \end{align} \tag{ 40 }$$

into two pieces, according to the decomposition of $\mu_2$ given by formula (23), that is

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \label{122A} \mu_2=\bigg[D\tilde{Q}_{\lambda} \left(\begin{array}{@{}l@{}} U \nonumber \\ 0 \end{array}\right)\bigg]_2 + f_2. \nonumber \end{align} \tag{ 41 }$$

By lemma 3.4 we have that $\frac{1}{|\lambda|^A} f \in \mathcal{H}^{\,p}$, for any p   >  2. Therefore, we apply (41) to (40) and we split the integral into I₁ and I₂, according to the order in (41).

Since, by assumption, w is an admissible point there exists a $\lambda_s$ fulfilling the inequality $\sup_{z \in \overline{\mathcal{D}}}\, {\rm Re}[\lambda_s(z-w){}^2] < -1/2.$

So, for $z \in \Gamma^{+}$ we get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber \left| |\lambda|^A {\rm e}^{\overline{\ln |\lambda|\lambda_s(z-w)^2}} \right|&=|\lambda|^A|{\rm e}^{\ln |\lambda|\,{\rm Re}[\lambda_s(z-w)^2]}| < |\lambda|^A {\rm e}^{-1/2\ln|\lambda|}=|\lambda|^{A-1/2} \nonumber \\ \left| |\lambda|^A {\rm e}^{\overline{\ln |\lambda|\lambda_s(z-w)^2}}\right| & < |\lambda|^{-\delta}, \nonumber \end{align} \tag{ 42 }$$

where we choose $-\delta=A-1/2 < 0$ (recall, A  <  1/2). Hence, we obtain

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle |I_2| &\leqslant \frac{1}{|\lambda|}\int_{\Gamma^{+}} \bigg||\lambda|^A{\rm e}^{\overline{\ln |\lambda|\lambda_s(z-w)^2}} \bigg(\frac{1}{|\lambda|^A}f_2\bigg)\bigg| \, {\rm d}|\bar{z}| \nonumber \\ & < \frac{1}{|\lambda|^{1+\delta}} \int_{\Gamma^{+}} \bigg|\frac{1}{|\lambda|^A}f_2\bigg| \, {\rm d}|\bar{z}|. \nonumber \end{align*}$$

Integrating with respect to the spectral parameter, we have for R  >  0 large enough

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \int_{|\lambda| > R} |I_2| {\rm d}\sigma_{\lambda} &\leqslant \int_{|\lambda| > R} \frac{1}{|\lambda|^{1+\delta}} \int_{\Gamma^{+}} \bigg|\frac{1}{|\lambda|^A}f_2\bigg| \, {\rm d}|\bar{z}| {\rm d}\sigma_{\lambda} \nonumber \\ & \leqslant \left\|\frac{1}{|\lambda|^{1+\delta}}\right\|_{L^q_{\lambda}} \left\|\int_{\Gamma^{+}} \bigg|\frac{1}{|\lambda|^A}f_2\bigg| \, {\rm d}|\bar{z}|\right\|_{L^{\,p}_{\lambda}}. \nonumber \end{align*}$$

Therefore, by lemma 3.4 the second norm is finite for p   >  2. We now pick q such that $q(1+\delta) > 2$, which is always possible given that $\delta > 0$. Hence, I₂ is in $L^1(\lambda:|\lambda| > R).$

Now, we look at I₁. By definition we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I_1=\frac{1}{2|\lambda|} \int_{\Gamma^{+}} {\rm e}^{\overline{\ln |\lambda|\lambda_s(z-w)^2}}\int_{\mathcal{O}} \frac{{\rm e}^{\ln |\lambda|\lambda_s(z_1-w)^2+i\,{\rm Im}(\lambda(z_1-w)^2)/2}}{\bar{z}-\bar{z}_1}Q_{21}(z_1) \,{\rm d}\sigma_{z_1}\,{\rm d}\bar{z}. \nonumber \end{align} \tag{ 43 }$$

Again, integrating against the spectral parameter we get:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_{|\lambda| > R} |I_1| {\rm d}\sigma_{\lambda} &\leqslant \int_{|\lambda| > R} \frac{1}{2|\lambda|}\int_{\Gamma^{+}} |\lambda|^{-\delta} \Bigg|\frac{1}{|\lambda|^A}\int_{\mathcal{O}} \frac{{\rm e}^{\ln |\lambda|\lambda_s(z_1-w)^2+i\,{\rm Im}(\lambda(z_1-w)^2)/2}}{\bar{z}-\bar{z}_1}Q_{21}(z_1) {\rm d}\sigma_{z_1}\Bigg|\, {\rm d}|\bar{z}|\,{\rm d}\sigma_{\lambda} \nonumber \\ & = \int_{\Gamma^{+}}\int_{|\lambda| > R} \frac{1}{2|\lambda|^{1+\delta}} \Bigg|\frac{1}{|\lambda|^A}\int_{\mathcal{O}} \frac{{\rm e}^{\ln |\lambda|\lambda_s(z_1-w)^2+i\,{\rm Im}(\lambda(z_1-w)^2)/2}}{\bar{z}-\bar{z}_1}Q_{21}(z_1) {\rm d}\sigma_{z_1}\Bigg|\, {\rm d}\sigma_{\lambda}\,{\rm d}|\bar{z} \nonumber \\ & \leqslant \Bigg|\Bigg|\frac{1}{2|\lambda|^{1+\delta}}\Bigg|\Bigg|_{L^q_{\lambda}} \int_{\Gamma^{+}} \Bigg|\Bigg|\frac{1}{|\lambda|^A}\int_{\mathcal{O}} \frac{{\rm e}^{\ln |\lambda|\lambda_s(z_1-w)^2+i\,{\rm Im}(\lambda(z_1-w)^2)/2}}{\bar{z}-\bar{z}_1}Q_{21}(z_1) {\rm d}\sigma_{z_1}\Bigg|\Bigg|_{L^{\,p}_{\lambda}} {\rm d}|\bar{z}|, \nonumber \end{align*}$$

where we use Fubini's theorem and Hölder inequality, with p   >  2 small enough so that the first norm is finite as in the computation of I₂.

Now, we can use lemma 3.6, given that we assume that our potential Q has support in O and it is in $L^{\infty}_{z}$, to obtain a constant C  >  0 depending only on the support of the potential and on a certain $\tilde{\delta} > 0$:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \int_{|\lambda| > R} |I_1| {\rm d}\sigma_{\lambda} \leqslant C \left\| \frac{1}{2|\lambda|^{1+\delta}}\right\|_{L^q_{\lambda}} \|Q_{21} \|_{L^{\infty}_{z}} \int_{\Gamma^{+}} \frac{1}{|\bar{z}-w|^{1-\tilde{\delta}}} \, {\rm d}|\bar{z}|. \nonumber \end{align*}$$

Given that the last integral is finite, we have $I_1 \in L^1(\lambda:|\lambda| > R)$ and the desired result follows.

Now we can present the proof of our main theorem, using the lemmas of the previous section while paying close attention to how $\mu$ and f  are defined.

Proof. Let us start by taking a look at the following term

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Hsc} \nonumber \frac{h(\lambda,w)}{|\lambda|}= \frac{1}{|\lambda|}\Bigg[&\int_{\Gamma^{+}} {\rm e}^{\overline{\ln|\lambda|\lambda_s(z-w)^2}}\mu_2(z){\rm d}\bar{z}\nonumber \\ &+\int_{\mathcal{O}\setminus\overline{\mathcal{D}}} {\rm e}^{\overline{{{\rm ln}}|\lambda|\lambda_s(z-w)^2}}{\rm e}^{-{\rm i}\,{\rm Im}(\lambda(z-w)^2)/2} q_{21}(z)\mu_1(z) {\rm d}\sigma_z\Bigg]. \nonumber \end{align} \tag{ 44 }$$

From (23) we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \mu=f+(I+D\tilde{Q}_{\lambda}) \left(\begin{array}{@{}c@{}} U \nonumber \\ 0 \end{array}\right),\nonumber \end{align*}$$

whereby f  is a solution of

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle f=-\Bigg(Mf + M(I+D\tilde{Q}_{\lambda}) \left(\begin{array}{@{}c@{}} U \nonumber \\ 0 \end{array}\right)\Bigg).\nonumber \end{align*}$$

This leads to $\newcommand{\e}{{\rm e}} \mu_1=-\left[Mf+ M(I+D\tilde{Q}_{\lambda}) \left(\begin{array}{@{}c@{}} U \\ 0 \end{array}\right)\right]_1+U.$ Therefore, by (44) and the definition of the operator T, we get

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \nonumber \frac{h(\lambda,w)}{|\lambda|}=\frac{1}{|\lambda|}&\int_{\Gamma^{+}} {\rm e}^{\overline{\ln|\lambda|\lambda_s(z-w)^2}}\mu_2(z){\rm d}\bar{z}-\frac{1}{|\lambda|}T\bigg(\big[Mf\big]_1\bigg)\nonumber \\&-\frac{1}{|\lambda|}T\bigg( \bigg[M(I+D\tilde{Q}_{\lambda}) \left(\begin{array}{@{}c@{}} U \nonumber \\ 0 \end{array}\right)\bigg]_1\bigg) + \frac{1}{|\lambda|}T[U] =:A+B+C+D. \nonumber \end{align} \tag{ 45 }$$

We need to study the terms $A,\,B,\,C,\,D$. By lemma 3.8, we have for p   <  2 and R large enough that

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle B,C \in L^{\,p}(\lambda:|\lambda|> R).\nonumber \end{align*}$$

From lemma 3.9, we obtain

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle A \in L^1(\lambda:|\lambda|> R).\nonumber \end{align*}$$

Hence, we just need to analyze the behavior of the last term.

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle T[U]&=\int_{\mathcal{O}\setminus\overline{\mathcal{D}}} {\rm e}^{\overline{{{\rm ln}}|\lambda|\lambda_s(z-w)^2}}{\rm e}^{-{\rm i}\,{\rm Im}(\lambda(z-w)^2)/2} q_{21}(z){\rm e}^{\ln|\lambda|\lambda_s(z-w)^2} {\rm d}\sigma_z \nonumber \\ &= \int_{\mathcal{O}\setminus\overline{\mathcal{D}}} {\rm e}^{\overline{{{\rm ln}}|\lambda|(\sqrt{\lambda_s}z-\sqrt{\lambda_s}w)^2}}{\rm e}^{-{\rm i}\,{\rm Im}(\lambda(z-w)^2)/2} q_{21}(z){\rm e}^{\ln|\lambda|(\sqrt{\lambda_s}z-\sqrt{\lambda_s}w)^2} {\rm d}\sigma_z \nonumber \\ &= \frac{1}{\lambda_s}\int_{\mathcal{O}\setminus \overline{\mathcal{D}}} {\rm e}^{\overline{{{\rm ln}}|\lambda|(z-\sqrt{\lambda_s}w)^2}}{\rm e}^{-{\rm i}\,{\rm Im}(\lambda/\sqrt{\lambda_s}(z-\sqrt{\lambda_s}w)^2)/2} q_{21}(z)\nonumber \\ & \hspace{2.5cm}{\rm e}^{\ln|\lambda|(\sqrt{\lambda_s}z-\sqrt{\lambda_s}w)^2}{\rm e}^{-{\rm i}\,{\rm Im}(\lambda(z-\sqrt{\lambda_s}w)^2)} {\rm e}^{{\rm i}\,{\rm Im}(\lambda(z-\sqrt{\lambda_s}w)^2)} {\rm d}\sigma_z, \nonumber \end{align*}$$

where we did a simple change of variables. We define

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \phi(z)={\rm e}^{-{\rm i}\,{\rm Im}(\lambda/\lambda_s(z-\sqrt{\lambda_s}w)^2)/2}{\rm e}^{{\rm i}\,{\rm Im}(\lambda(z-\sqrt{\lambda_s}w)^2)} {\rm e}^{\overline{{{\rm ln}}|\lambda|(z-\sqrt{\lambda_s}w)^2}} q_{21}(z/\sqrt{\lambda_s}).\nonumber \end{align*}$$

Given that the conditions of lemma 3.7 are fulfilled, we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T[U]=\frac{1}{\lambda_s}\Bigg[\frac{2\pi}{|\lambda|}\phi(\sqrt{\lambda_s}w)+R_{\sqrt{\lambda_s}w}(\lambda)\Bigg], \nonumber \end{align} \tag{ 46 }$$

which by substitution implies

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \frac{1}{|\lambda|}T[U]=\frac{1}{\lambda_s}\frac{2\pi}{|\lambda|}q_{21}(w)+\frac{1}{\lambda_s}|\lambda|^{-1}R_{\sqrt{\lambda_s}w}(\lambda)=:D_1+D_2.\nonumber \end{align*}$$

By lemma 3.7, we have $D_2 \in L^1(\lambda:|\lambda|> R)$.

So finally we are ready to evaluate the left-hand side of (32):

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \lim_{R \to \infty} \int_{R < |\lambda|< 2R} |\lambda|^{-1} h(\lambda,w) {\rm d}\sigma_{\lambda} &= \lim_{R \to \infty} \int_{R < |\lambda|< 2R} \frac{2\pi}{\lambda_s}|\lambda|^{-2}q_{21}(w) {\rm d}\sigma_{\lambda} \nonumber \\ &= q_{21}(w)\frac{4\pi^2}{\lambda_s} \lim_{R \to \infty} \int_{R}^{2R} r^{-1} {\rm d}r \nonumber \\ &= q_{21}(w)\frac{4\pi^2}{\lambda_s} \lim_{R \to \infty} \ln r\Big|_{R}^{2R} = q_{21}(w)\frac{4\pi^2\ln 2}{\lambda_s}. \nonumber \end{align*}$$

From this we get the desired asymptotic:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle q_{21}(w)=\frac{\lambda_s}{4\pi^2\ln 2} \lim_{R \to \infty} \int_{R < |\lambda|< 2R} |\lambda|^{-1} h(\lambda,w) {\rm d}\sigma_{\lambda}.\nonumber \end{align*}$$