Approximate Quantum and Acoustic Cloaking

Abstract

For any E\ge 0 , we construct a sequence of bounded potentials V^E_{n},\, n\in\Bbb N , supported in {an annular region B_{out}\setminus B_{inn}\subset\Bbb 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^E_{ n} 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.