Approximate quantum and acoustic cloaking

Abstract

At any energy E > 0, we construct a sequence of bounded potentials $V_n^E,n\in \mathbb{N}$, supported in an annular region $B_{out}\setminus B_{inn}$ in three-space, which act as approximate cloaks for solutions of Schrödinger's equation: For any potential $V_0\in L^{\infty }(B_{inn})$ such that E is not a Neumann eigenvalue of $-\Delta +V_0$ in $B_{inn}$, the scattering amplitudes $a_{V_0+V_n^E}(E,\theta ,\omega )\to 0$ as $n\to \infty$. The $V_n^E$ 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 $B_{inn}$. We derive the $V_n^E$ 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.

2 color figures

References in corpus (6)

• Optical design of reflectionless complex media by finite embedded coordinate transformations
• Confirmation of Cylindrical Perfect Invisibility Cloak Using Fourier-Bessel Analysis
• Electromagnetic wormholes and virtual magnetic monopoles
• Design and analytically full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations
• Electromagnetic wave manipulation using layered systems
• Comment on "Scattering Theory Derivation of a 3D Acoustic Cloaking Shell"

Cited by in corpus (1)

• Hierarchical models in statistical inverse problems and the Mumford--Shah functional