2008/12/09 by Allan Greenleaf, Greenleaf, Allan, Yaroslav Kurylev +5
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.0812.1706
2 color figures
arxiv created 2008/12/09 · arxiv updated 2009/12/01
At any energy E > 0, we construct a sequence of bounded potentials VEn, n∈\N, supported in an annular region Bout∖ Binn in three-space, which act as approximate cloaks for solutions of Schrödinger's equation: For any potential V0∈ L^∞(Binn) such that E is not a Neumann eigenvalue of -Δ+V0 in Binn, the scattering amplitudes aV0+VnE(E,θ,ω)→ 0 as n→∞. The VEn 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 \it almost trapped states concentrated in Binn. We derive the VnE 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.