近似量子和声学隐身
Approximate quantum and acoustic cloaking
摘要：
在任意能量E > 0时，我们构造了一个支撑在三维空间中的环形区域$B_{out}\setminus B_{inn}$中的有界势$V^E_{n}, n\in\N$序列，作为薛定谔方程解的近似伪装：对于任意的势$V_0\in L^\infty(B_{inn})$，使得E不是$B_{inn}$中$-Δ+V_0$的Neumann本征值，散射振幅$a_{V_0+V_n^E}(E,θ,ω)\to 0$为$n\to\infty$ .因此，$V^E_{n}$不仅形成近似透明势族，而且在量子力学中起到近似隐形斗篷的作用。另一方面，对于接近内部本征值的$E$，共振发展并且存在集中在$B_{inn}$中的{ \it几乎陷阱态}。我们从奇异的，各向异性的变换光学为基础的斗篷通过去各向异性的过程，我们称之为\emph{isotropic transformation optics}的$V_n^E$。该技术使用截断，逆均匀化和谱理论产生非奇异，各向同性的近似斗篷。作为一个中间步骤，我们还获得了近似的隐形一般类的方程，包括声学方程。
英文摘要：
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 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.
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