Approximate Quantum and Acoustic Cloaking

  • Allan Greenleaf

    University of Rochester, USA
  • Yaroslav Kurylev

    University College London, UK
  • Matti Lassas

    University of Helsinki, Finland
  • Gunther Uhlmann

    University of Washington, Seattle, United States


For any , we construct a sequence of bounded potentials , supported in {an annular region ,} which act as approximate cloaks for solutions of Schrödinger's equation at energy : For any potential {such that is not a Neumann eigenvalue of in }, the scattering amplitudes as . The 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 close to interior eigenvalues, resonances develop and there exist almost trapped states concentrated in .} We derive the 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.

Cite this article

Allan Greenleaf, Yaroslav Kurylev, Matti Lassas, Gunther Uhlmann, Approximate Quantum and Acoustic Cloaking. J. Spectr. Theory 1 (2011), no. 1, pp. 27–80

DOI 10.4171/JST/2