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The X-ray transform on asymptotically conic spaces

András Vasy and Evangelie Zachos
Vol. 6 (2024), No. 3, 693–730
DOI: 10.2140/paa.2024.6.693
Abstract

In this paper, partly based on Zachos’ PhD thesis, we show that the geodesic X-ray transform is stably invertible near infinity on a class of asymptotically conic manifolds which includes perturbations of Euclidean space. In particular certain kinds of conjugate points are allowed. Further, under a global convex foliation condition, the transform is globally invertible.

The key analytic tool, beyond the approach introduced by Uhlmann and Vasy, is the introduction of a new pseudodifferential operator algebra, which we name the 1-cusp algebra, and its semiclassical version.

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Keywords
X-ray transform, inverse problems, microlocal analysis, 1-cusp pseudodifferential algebra, asymptotically conic manifolds
Mathematical Subject Classification
Primary: 35S05, 53C65
Milestones
Received: 15 January 2023
Revised: 12 December 2023
Accepted: 18 March 2024
Published: 1 October 2024
Authors
András Vasy
Department of Mathematics
Stanford University
Stanford, CA
United States
Evangelie Zachos
Department of Mathematics
Stanford University
Stanford, CA
United States
© 2024 MSP (Mathematical Sciences Publishers).