# 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

## Abstract

For any $E≥0$, we construct a sequence of bounded potentials $V_{n},n∈N$, supported in {an annular region $B_{out}∖B_{inn}⊂R_{3}$,} which act as approximate cloaks for solutions of Schrödinger's equation at energy $E$: For any potential $V_{0}∈L_{∞}(B_{inn})$ {such that $E$ is not a Neumann eigenvalue of $−Δ+V_{0}$ in $B_{inn}$}, the scattering amplitudes $a_{V_{0}+V_{n}}(E,θ,ω)→0$ as $n→∞$. The $V_{n}$ 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 $E$ close to interior eigenvalues, resonances develop and there exist *almost trapped states* concentrated in $B_{inn}$.} We derive the $V_{n}$ 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