# UCL Discovery

## Approximate quantum and acoustic cloaking

Greenleaf, A and Kurylev, Y and Lassas, M and Uhlmann, G (2008) Approximate quantum and acoustic cloaking.

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## Abstract

At any energy E > 0, we construct a sequence of bounded potentials $V^E_{n}, n\in\N$, supported in an annular region $B_{out}\setminus B_{inn}$ in three-space, which act as approximate cloaks for solutions of Schr\"odinger'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 $V^E_{n}$ 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 {\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.

Type: Article Approximate quantum and acoustic cloaking 2 color figures UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics

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