Approximate Quantum and Acoustic Cloaking

Abstract

For any $E\ge 0$, we construct a sequence of bounded potentials $V_n^E,n\in \mathbb{N}$, supported in {an annular region $B_{out}\setminus B_{inn}\subset {\mathbb{R}}^3$,} which act as approximate cloaks for solutions of Schrödinger's equation at energy $E$: For any potential $V_0\in L^{\infty }(B_{inn})$ {such that $E$ is not a Neumann eigenvalue of $-\Delta +V_0$ in $B_{inn}$}, the scattering amplitudes $a_{V_0+V_n^E}(E,\theta ,\omega )\to 0$ as $n\to \infty$. The $V_n^E$ thus not only form a family of approximately transparent potentials, but also function as approximate invisibility cloaks in quantum mechanics. {On the other hand, for $E$ close to interior eigenvalues, resonances develop and there exist almost trapped states concentrated in $B_{inn}$.} We derive the $V_n^E$ from singular, anisotropic transformation optics-based cloaks by a de-anisotropization procedure, which we call \emph{isotropic transformation optics}. This technique uses truncation, inverse homogenization and spectral theory to produce nonsingular, isotropic approximate cloaks. As an intermediate step, we also obtain approximate cloaking for a general class of equations including the acoustic equation.