Approximate quantum and acoustic cloaking

For any E >= 0, we construct a sequence of bounded potentials V-n(E), n is an element of N, supported in an annular region B-out \ B-in subset of R-3, which act as approximate cloaks for solutions of Schrodinger's equation at energy E: for any potential V-0 is an element of L-infinity(B-in) such that E is not a Neumann eigenvalue of -Delta + V-0 in B-in, the scattering amplitudes alpha(V0)+V-n(E) (E, theta, omega) -> 0 as n -> infinity. The V-n(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-in. We derive the V-n(E) n from singular, anisotropic transformation optics-based cloaks by a de-anisotropization procedure, which we call 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.