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The Calder\'on problem for the Schr\'odinger equation in transversally anisotropic geometries with partial data

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Title: The Calder\'on problem for the Schr\'odinger equation in transversally anisotropic geometries with partial data
Authors: Lin, Yi-Hsuan, Nakamura, Gen, Zimmermann, Philipp
Publication Year: 2024
Collection: Mathematics
Subject Terms: Mathematics - Analysis of PDEs
More Details: We study the partial data Calder\'on problem for the anisotropic Schr\"{o}dinger equation \begin{equation} \label{eq: a1} (-\Delta_{\widetilde{g}}+V)u=0\text{ in }\Omega\times (0,\infty), \end{equation} where $\Omega\subset\mathbb{R}^n$ 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 $\Gamma\times \{0\}$ with $\Gamma\Subset \Omega$. Our approach can be divided into three steps: Step 1. Boundary determination. We establish a novel boundary determination to identify $(g,V)$ on $\Gamma$ with help of suitable approximate solutions for the Schr\"odinger equation with inhomogeneous Neumann boundary condition. Step 2. Relation to a nonlocal elliptic inverse problem. We relate inverse problems for the Schr\"odinger equation with the nonlocal elliptic equation \begin{equation} \label{eq: a2} (-\Delta_g+V)^{1/2}v=f\text{ in }\Omega, \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-\Delta_g+V)w=F\text{ in }\Omega\times (0,\infty), \end{equation} where the measurement operator is also the source-to-solution map. We can finally determine $(g,V)$ on $\Omega\setminus\Gamma$ by solving the inverse problem for the wave equation.
Comment: 54 pages. All comments are welcome
Document Type: Working Paper
Access URL: http://arxiv.org/abs/2408.08298
Accession Number: edsarx.2408.08298
Database: arXiv
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  Data: The Calder\'on problem for the Schr\'odinger equation in transversally anisotropic geometries with partial data
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  Data: <searchLink fieldCode="AR" term="%22Lin%2C+Yi-Hsuan%22">Lin, Yi-Hsuan</searchLink><br /><searchLink fieldCode="AR" term="%22Nakamura%2C+Gen%22">Nakamura, Gen</searchLink><br /><searchLink fieldCode="AR" term="%22Zimmermann%2C+Philipp%22">Zimmermann, Philipp</searchLink>
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  Data: 2024
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  Data: Mathematics
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  Label: Description
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  Data: We study the partial data Calder\'on problem for the anisotropic Schr\"{o}dinger equation \begin{equation} \label{eq: a1} (-\Delta_{\widetilde{g}}+V)u=0\text{ in }\Omega\times (0,\infty), \end{equation} where $\Omega\subset\mathbb{R}^n$ 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 $\Gamma\times \{0\}$ with $\Gamma\Subset \Omega$. Our approach can be divided into three steps: Step 1. Boundary determination. We establish a novel boundary determination to identify $(g,V)$ on $\Gamma$ with help of suitable approximate solutions for the Schr\"odinger equation with inhomogeneous Neumann boundary condition. Step 2. Relation to a nonlocal elliptic inverse problem. We relate inverse problems for the Schr\"odinger equation with the nonlocal elliptic equation \begin{equation} \label{eq: a2} (-\Delta_g+V)^{1/2}v=f\text{ in }\Omega, \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-\Delta_g+V)w=F\text{ in }\Omega\times (0,\infty), \end{equation} where the measurement operator is also the source-to-solution map. We can finally determine $(g,V)$ on $\Omega\setminus\Gamma$ by solving the inverse problem for the wave equation.<br />Comment: 54 pages. All comments are welcome
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RecordInfo BibRecord:
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    Subjects:
      – SubjectFull: Mathematics - Analysis of PDEs
        Type: general
    Titles:
      – TitleFull: The Calder\'on problem for the Schr\'odinger equation in transversally anisotropic geometries with partial data
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            NameFull: Lin, Yi-Hsuan
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            NameFull: Nakamura, Gen
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            NameFull: Zimmermann, Philipp
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              Y: 2024
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