and established as a rigorous regularization strategy in \[[@Knudsen2009]\]. Since this method only applies to real-valued conductivities, consider $\gamma=\sigma$. In this method, equation is first transformed into the Schrödinger equation $[-\Delta + q(z)]\tilde{u}(z) = 0$, for $z\in\Omega\subset{{\mathbb R}}^2$ via the change of variables $\tilde{u}=\sigma^{1/2}u$, $q(z)=\sigma^{-1/2}\Delta\sigma^{1/2}$. Assume $\sigma$ is the constant $1$ in a neighborhood of the boundary, and extend the conductivity from $\Omega$ to the full plane as $\sigma(z)\equiv 1$ for $z\in{{\mathbb R}}^2\setminus \Omega$. Note that the the assumption that $\sigma$ is constant near the boundary can be dropped, and the DN map extended through analytic continuation, so the method then applies on a slightly larger domain. See \[[@SiltanenTamminen2012], [@Nachman1996]\] for further details on this approach.

Nachman proved the existence of [[unique]{}]{} CGO solutions $\psi_s(z,k)$ to the Schrödinger equation $$\label{eq:schro}
[-\Delta + q(z)]\psi_s(z,k) = 0,\qquad z\in{{\mathbb C}},\;\;k\in{{\mathbb C}}\setminus0,$$ where [[$e^{-ikz}\psi_s(z.k)-1 \in W^{1,p}({{\mathbb R}}^2)$, $p>2$, i.e. $\psi_s\sim e^{ikz}$ for large $|k|$ or $|z|$.]{}]{} Here $k$ is a complex number, and $kz$ is complex multiplication. The CGO solutions $\mu(z,k)\equiv e^{-ikz}\psi_s(z,k)$ satisfy a D-bar equation in the transform variable $k$: $$\label{eq:Nach_dbark}
{\bar{\partial}}_k\mu(z,k)=\frac{1}{4\pi\bar{k}}{{\mathbf{t}}}(k)e(z,-k)\overline{\mu(z,k)},$$ where $e(z,k)\equiv e^{i(kz+\bar{k}\bar{z})}=e^{2i\Re(zk)}$ and ${{\mathbf{t}}}(k)$ denotes the scattering data, which is related to the DN map via $$\label{eq:Nach_scatBndry}
{{\mathbf{t}}}(k)=\int_{\operatorname{\partial\Omega}} e^{i\bar{k}\bar{z}}(\Lambda_\sigma-\Lambda_1)\psi_s(z,k)\;dz,$$ where 