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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.08298 |
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| _version_ | 1866916359035682816 |
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| author | Lin, Yi-Hsuan Nakamura, Gen Zimmermann, Philipp |
| author_facet | Lin, Yi-Hsuan Nakamura, Gen Zimmermann, Philipp |
| contents | We study the partial data Calderón problem for the anisotropic Schrödinger equation
\begin{equation}
\label{eq: a1}
(-Δ_{\widetilde{g}}+V)u=0\text{ in }Ω\times (0,\infty),
\end{equation}
where $Ω\subset\mathbb{R}^n$ is a bounded smooth domain, $\widetilde{g}=g_{ij}(x)dx^{i}\otimes dx^j+dy\otimes dy$ and $V$ is translationally invariant in the $y$ direction. Our goal is to recover both the metric $g$ and the potential $V$ from the (partial) Neumann-to-Dirichlet (ND) map on $Γ\times \{0\}$ with $Γ\Subset Ω$. Our approach can be divided into three steps:
Step 1. Boundary determination. We establish a novel boundary determination to identify $(g,V)$ on $Γ$ with help of suitable approximate solutions for the Schrödinger equation with inhomogeneous Neumann boundary condition.
Step 2. Relation to a nonlocal elliptic inverse problem. We relate inverse problems for the Schrödinger equation with the nonlocal elliptic equation
\begin{equation}
\label{eq: a2}
(-Δ_g+V)^{1/2}v=f\text{ in }Ω,
\end{equation}
via the Caffarelli--Silvestre type extension, where the measurements are encoded in the source-to-solution map. The nonlocality of this inverse problem allows us to recover the associated heat kernel.
Step 3. Reduction to an inverse problem for a wave equation. Combining the knowledge of the heat kernel with the Kannai type transmutation formula, we transfer the inverse problem for the nonlocal equation to an inverse problem for the wave equation
\begin{equation}
\label{eq: a3}
(\partial_t^2-Δ_g+V)w=F\text{ in }Ω\times (0,\infty),
\end{equation}
where the measurement operator is also the source-to-solution map. We can finally determine $(g,V)$ on $Ω\setminusΓ$ by solving the inverse problem for the wave equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Calderón problem for the Schrödinger equation in transversally anisotropic geometries with partial data Lin, Yi-Hsuan Nakamura, Gen Zimmermann, Philipp Analysis of PDEs We study the partial data Calderón problem for the anisotropic Schrödinger equation \begin{equation} \label{eq: a1} (-Δ_{\widetilde{g}}+V)u=0\text{ in }Ω\times (0,\infty), \end{equation} where $Ω\subset\mathbb{R}^n$ is a bounded smooth domain, $\widetilde{g}=g_{ij}(x)dx^{i}\otimes dx^j+dy\otimes dy$ and $V$ is translationally invariant in the $y$ direction. Our goal is to recover both the metric $g$ and the potential $V$ from the (partial) Neumann-to-Dirichlet (ND) map on $Γ\times \{0\}$ with $Γ\Subset Ω$. Our approach can be divided into three steps: Step 1. Boundary determination. We establish a novel boundary determination to identify $(g,V)$ on $Γ$ with help of suitable approximate solutions for the Schrödinger equation with inhomogeneous Neumann boundary condition. Step 2. Relation to a nonlocal elliptic inverse problem. We relate inverse problems for the Schrödinger equation with the nonlocal elliptic equation \begin{equation} \label{eq: a2} (-Δ_g+V)^{1/2}v=f\text{ in }Ω, \end{equation} via the Caffarelli--Silvestre type extension, where the measurements are encoded in the source-to-solution map. The nonlocality of this inverse problem allows us to recover the associated heat kernel. Step 3. Reduction to an inverse problem for a wave equation. Combining the knowledge of the heat kernel with the Kannai type transmutation formula, we transfer the inverse problem for the nonlocal equation to an inverse problem for the wave equation \begin{equation} \label{eq: a3} (\partial_t^2-Δ_g+V)w=F\text{ in }Ω\times (0,\infty), \end{equation} where the measurement operator is also the source-to-solution map. We can finally determine $(g,V)$ on $Ω\setminusΓ$ by solving the inverse problem for the wave equation. |
| title | The Calderón problem for the Schrödinger equation in transversally anisotropic geometries with partial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.08298 |