arXiv subjects

Shubham R. Jathar

Publications and source records attributed to Shubham R. Jathar.

8 recordsLinked to original sources

arXiv2026

The matrix weighted real-analytic double fibration transforms

We show that the real-analytic matrix-weighted double fibration transform determines the analytic wavefront set of a vector-valued function. We apply this result to show that the matrix weighted ray transform is injective on a two-dimensional, non-trapping, real-analytic Riemannian manifold with strictly convex boundary. Additionally, we show that a real-analytic Higgs field can be uniquely determined from the nonabelian ray transform on real-analytic Riemannian manifolds of any dimension with a strictly convex boundary point.

math.AP↗

arXiv2024

Loop group factorization method for the magnetic and thermostatic nonabelian ray transforms

We study the injectivity of the matrix attenuated and nonabelian ray transforms on compact surfaces with boundary for nontrapping $λ$-geodesic flows and the general linear group of invertible complex matrices. We generalize the loop group factorization argument of Paternain and Salo to reduce to the setting of the unitary group when $λ$ has the vertical Fourier degree at most $2$. This covers the magnetic and thermostatic flows as special cases. Our article settles the general injectivity question of the nonabelian ray transform for simple magnetic flows in combination with an earlier result by Ainsworth. We stress that the injectivity question in the unitary case for simple Gaussian thermostats remains open. Furthermore, we observe that the loop group argument does not apply when $λ$ has higher Fourier modes.

math.DG↗

arXiv2024

Weighted Divergent Beam Ray Transform: Reconstruction, Unique continuation and Stability

In this article, we establish that any symmetric $m$-tensor field can be recovered pointwise from partial data of the $k$-th weighted divergent ray transform for any $k \in \mathbb{Z}^{+} \cup\{0\}$. Using the unique continuation property of the fractional Laplacian, we further prove the unique continuation of the fractional divergent beam ray transform for both vector fields and symmetric 2-tensor fields. Additionally, we derive explicit reconstruction formulas and stability results for vector fields and symmetric 2-tensor fields in terms of fractional divergent beam ray transform data. Finally, we conclude by proving a unique continuation result for the divergent beam ray transform for functions.

math.AP↗

arXiv2024

Survey on Broken Ray Transforms

We survey recent developments in the theory and applications of the broken ray transforms. Furthermore, we discuss some open problems.

math.DG↗

arXiv2024

Inverse Problems for Twisted Geodesic Flows

The article surveys inverse problems related to the twisted geodesic flows on Riemannian manifolds with boundary, focusing on the generalized ray transforms, tensor tomography, and rigidity problems. The twisted geodesic flow generalizes the usual geodesic flow, allowing to model richer dynamics.

math.DG↗

arXiv2024

Normal operators for momentum ray transforms, I: The inversion formula

The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on $\mathbb R^n$ over lines with the weight $t^k$, $I_m^kf(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,\mathrm{d}t$. We compute the normal operator $N_m^k=(I_m^k){}^*I_m^k$ and present an inversion formula recovering a rank $m$ tensor field $f$ from the data $(N_m^0f,\dots,N_m^mf)$.

math.AP↗

arXiv2024

Normal operators for momentum ray transforms, II: Saint Venant operator

The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on ${\mathbb R}^n$ over lines with the weight $t^k$, $I_m^kf(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,\mathrm{d}t$. Let $N^k_m=(I^k_m)^*I^k_m$ be the normal operator of $I_m^k$. To what extent is a symmetric $m$-tensor field $f$ determined by the data $(N_m^0f,\dots,N_m^rf)$ given for some $0\le r\le m$? The Saint Venant operator $W^r_m$ is a linear differential operator of order $m-r$ with constant coefficients on the space of symmetric $m$-tensor fields. We derive an explicit formula expressing $W^r_mf$ in terms of $(N_m^0f,\dots,N_m^rf)$. The tensor field $W^r_mf$ represents the full local information on $f$ that can be extracted from the data $(N_m^0f,\dots,N_m^rf)$.

math.AP↗

arXiv2024

Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle

We prove a uniqueness result for the broken ray transform acting on the sums of functions and $1$-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.

math.DG↗