Approximate quantum and acoustic cloaking

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dc.creatorGreenleaf, Allan
dc.creatorKurylev, Yaroslav
dc.creatorLassas, Matti
dc.creatorUhlmann, Gunther
dc.date2008-12-09
dc.date.accessioned2026-07-07T12:10:50Z
dc.date.available2026-07-07T12:10:50Z
dc.descriptionAt 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 Schrödinger's equation: For any potential $V_0\in L^\infty(B_{inn})$ such that E is not a Neumann eigenvalue of $-Δ+V_0$ in $B_{inn}$, the scattering amplitudes $a_{V_0+V_n^E}(E,θ,ω)\to 0$ 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.
dc.description2 color figures
dc.identifierhttps://arxiv.org/abs/0812.1706
dc.identifierhttp://arxiv.org/abs/0812.1706
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/210047
dc.subjectMathematical Physics
dc.subjectAnalysis of PDEs
dc.titleApproximate quantum and acoustic cloaking
dc.typetext

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