Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators
Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators
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DOI：
10.3934/ipi.2019011
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发表时间：
2017-12
影响因子：
1.3
通讯作者：
Xinlin Cao;Yi-Hsuan Lin;Hongyu Liu
中科院分区：
文献类型：
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作者：
Xinlin Cao;Yi-Hsuan Lin;Hongyu Liu
设$A\in\mathrm{Sym}（n\times n）$是一个椭圆2-张量。考虑各向异性分数阶Schr\“odinger算子$\mathscr{L}_A^s+q$，其中$\mathscr{L}_A^s：=（-\nabla\cdot（A（x）\nabla））^s$，$s\in（0，1）$和$q\in L^\infty$。我们关注的同时恢复$q$和可能嵌入的软或硬障碍内$q$的外部Dirichlet到诺伊曼（DtN）映射外的有界域$\Omega$与$\mathscr{L}_A^s+q$。结果表明，一个单一的测量可以唯一地确定嵌入的障碍，独立于周围的潜在的q$。如果允许多次测量，那么周围的电势q也可以被唯一地恢复。这些是令人惊讶的发现，因为在当地的情况下，即$s=1$，障碍物恢复由一个单一的测量和周围的潜在的多个测量的同时恢复是长期存在的问题，仍然在文献中公开。我们对非局部反问题的讨论主要是基于各向异性分数阶Schr odinger算子的强唯一性和Runge逼近性质.
Let $A\in\mathrm{Sym}(n\times n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schr\"odinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s:=(-\nabla\cdot(A(x)\nabla))^s$, $s\in (0, 1)$ and $q\in L^\infty$. We are concerned with the simultaneous recovery of $q$ and possibly embedded soft or hard obstacles inside $q$ by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain $\Omega$ associated with $\mathscr{L}_A^s+q$. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential $q$. If multiple measurements are allowed, then the surrounding potential $q$ can also be uniquely recovered. These are surprising findings since in the local case, namely $s=1$, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are longstanding problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schr\"odinger operators.