# 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 $E\ge 0$, we construct a sequence of bounded potentials $V^E_{n},\, n\in\Bbb N$, supported in {an annular region $B_{out}\setminus B_{inn}\subset\Bbb R ^3$,} which act as approximate cloaks for solutions of Schrödinger's equation at energy $E$: 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 $V^E_{ n}$ 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 $E$ close to interior eigenvalues, resonances develop and there exist *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.

## 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