Approximate Quantum and Acoustic Cloaking

Allan Greenleaf, Yaroslav Kurylev, Matti Lassas, Gunther Uhlmann
2011 Journal of Spectral Theory  
For any E 0, we construct a sequence of bounded potentials V E n ; n 2 N, supported in an annular region B out n B in R 3 , which act as approximate cloaks for solutions of Schrödinger's equation at energy E: for any potential V 0 2 L 1 .B in / such that E is not a Neumann eigenvalue of C V 0 in B in , the scattering amplitudes a V 0 CV E n .E; Â; !/ ! 0 as n ! 1. The V E n thus not only form a family of approximately transparent potentials, but also function as approximate invisibility cloaks
more » ... n 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 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. Mathematics Subject Classification (2010). Primary 35R30; Secondary 35Q60, 35Q40.
doi:10.4171/jst/2 fatcat:yx5un5k64bdcfgt6h5h4moujp4