Global counterexamples to uniqueness for a Calder\'on problem with $C^k$ conductivities

Let $\Omega \subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $\gamma$ be a smooth conductivity in $\overline{ \Omega}$. Consider a non-zero frequency $\lambda_0$ which does not belong to the Dirichlet spectrum of $L_\gamma = -{ \rm div} (\gamma \nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(\gamma_1, \gamma_2)$ on $\overline{ \Omega}$, which are close to $\gamma$ such that the associated DN maps at frequency $\lambda_0$ satisfy \begin{ equation*} \Lambda_{ \gamma_1,\lambda_0} = \Lambda_{ \gamma_2,\lambda_0}. \end{ equation*}

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