JournalsjstVol. 1, No. 1pp. 27–80

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
Approximate Quantum and Acoustic Cloaking cover
Download PDF


For any E0E\ge 0, we construct a sequence of bounded potentials VnE,nNV^E_{n},\, n\in\Bbb N, supported in {an annular region BoutBinnR3B_{out}\setminus B_{inn}\subset\Bbb R ^3,} which act as approximate cloaks for solutions of Schrödinger's equation at energy EE: For any potential V0L(Binn)V_0\in L^\infty(B_{inn}) {such that EE is not a Neumann eigenvalue of Δ+V0-\Delta+V_0 in BinnB_{inn}}, the scattering amplitudes aV0+VnE(E,θ,ω)0a_{V_0+V_n^E}(E,\theta,\omega)\to 0 as nn\to\infty. The VnEV^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 EE close to interior eigenvalues, resonances develop and there exist almost trapped states concentrated in BinnB_{inn}.} We derive the VnEV_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.

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