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We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.
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Injectivity of the Heisenberg X-ray transform
We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier transform, we prove an operator-valued incarnation of the Fourier Slice Theorem, and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the family of taming metrics $$g_\epsilon$$ approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all $$\epsilon > 0$$. This result gives a concrete example of an injective X-ray transform in a geometry with an abundance of conjugate points.
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- Award ID(s):
- 1814104
- PAR ID:
- 10468491
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Functional Analysis
- Volume:
- 280
- Issue:
- 5
- ISSN:
- 0022-1236
- Page Range / eLocation ID:
- 108886
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jfa.2020.108886
