Document Zbl 1452.35255

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Funct. Anal.</a> <a href="/?q=in%3A448487" title="Articles in this Issue">279, No. 1, Article ID 108505, 42 p. (2020)</a>. </div> <div class="abstract">In this interesting paper under review, the authors show the global uniqueness in an inverse problem for the fractional Schrödinger equation with a single measurement and provide a reconstruction algorithm. <br class="zbmathjax-paragraph">More precisely, let $\Omega\subset \mathbb{R}^n$ be a bounded open set and $\Omega_e=\mathbb{R}^n\setminus \overline{\Omega}$ be the exterior domain. Let $\Lambda_q$ be the Dirichlet-to-Neumann map given by \[\Lambda_q: f\mapsto (-\Delta)^s u \big|_{\Omega_e},\] where $u$ and $f$ are related through the fractional Schrödinger equation \[ ((-\Delta)^s +q) u=0 \text{ in } \Omega, \quad u=f \text{ in } \Omega_e,\] with the potential $q$ chosen such that zero is not an eigenvalue. Assume that either $s\in [\frac{1}{4},1)$ and $q\in L^\infty(\Omega)$ or $q\in C^0(\overline{\Omega})$. The authors show that, given any nonempty open sets $W_1, W_2\subset \Omega_e$ with $\overline{\Omega}\cap W_1=\emptyset$, and given any nonzero function $f$ supported in $W_1$, the potential $q$ is uniquely determined by the single function $f$ and the single measurement $\Lambda_q(f)\big|_{W_2}$. Moreover, an explicit reconstruction procedure to recover the potential $q$ from $f$ and $\Lambda_q(f)$ is given. Note that the previous work [<span class="zbmathjax-textit">T. Ghosh</span> et al., Anal. PDE 13, No. 2, 455–475 (2020; <a href="/1439.35530">Zbl 1439.35530</a>)] considers infinitely many measurements, where one knows $\Lambda_q(f)\big|_{W_2}$ for all $f\in C_c^\infty(W_1)$.<br class="zbmathjax-paragraph">The proof is based on the strong approximation properties for the fractional Schrödinger equation and a unique continuation principle from sets of measure zero.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A15922">Wenhui Shi (Melbourne)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7190455">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7190455">Cited in <strong>57</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R30" title="MSC2020">35R30</a> </td> <td class="space"> Inverse problems for PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R11" title="MSC2020">35R11</a> </td> <td class="space"> Fractional partial differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B60" title="MSC2020">35B60</a> </td> <td class="space"> Continuation and prolongation of solutions to PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+Laplacian">fractional Laplacian</a>; <a href="/?q=ut%3Aunique+continuation">unique continuation</a>; <a href="/?q=ut%3Asingle+measurement">single measurement</a>; <a href="/?q=ut%3Afractional+Schr%C3%B6dinger+equation">fractional Schrödinger equation</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1439.35530">Zbl 1439.35530</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1452.35255" data-ciurl="/ci/07190455" data-biburl="/bibtex/07190455.bib" data-amsurl="/amsrefs/07190455.bib" data-xmlurl="/xml/07190455.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07190455.pdf" title="Zbl 1452.35255 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jfa.2020.108505" aria-label="DOI for “Uniqueness and reconstruction for the fractional Calderón problem with a single measurement”" title="10.1016/j.jfa.2020.108505">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1801.04449"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adolfsson, V.; Escauriaza, L., $ C^{1 , \alpha}$ domains and unique continuation at the boundary, Commun. 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