arXiv subjects

Yuzhou Joey Zou

Publications and source records attributed to Yuzhou Joey Zou.

5 recordsLinked to original sources

arXiv2026

The asymptotic structure of forward scattering

Perturbed plane waves are fundamental objects in scattering theory on Euclidean space and asymptotically Euclidean spaces. In this paper, we investigate the structure of perturbed plane waves in the $\textit{forward}$ direction, in which the outgoing spherical wave is typically singular and conjoined to the incoming plane wave. Melrose & Zworski provided a microlocal description (in the more general setting of asymptotically conic manifolds) using their notion of Lagrangian distributions associated to pairs of intersecting Legendrian submanifolds, on the way to proving that the S-matrix is an FIO. Here, we revisit the problem in the asymptotically Euclidean case, for which the oscillatory integrals used by Melrose--Zworski attain their most complicated form (relative to the more general asymptotically conic case). We seek a more elementary description in terms of physical-space asymptotics. These are specified using a two-faced compactification $X\hookleftarrow \mathbb{R}^d$, with one face for each asymptotic regime. We prove full polyhomogeneity. A transport equation arises as a model problem at the main face (`bf'). The quantum inverted harmonic oscillator arises as a model problem at the front face (`ff').

math.AP↗

arXiv2026

Tensor Tomography on Asymptotically Hyperbolic Surfaces

We initiate a study of the inversion of the geodesic X-ray transform $I_m$ over symmetric $m$-tensor fields on asymptotically hyperbolic surfaces. This operator has a non-trivial kernel whenever $m\ge 1$. To propose a gauge representative to be reconstructed from X-ray data, we first prove a "tt-potential-conformal" decomposition theorem for $m$-tensor fields (where "tt" stands for transverse traceless), previously used in integral geometry on compact Riemannian manifolds with boundary in Sharafudtinov, 2007, Dairbekov and Sharafutdinov, 2011. The proof is based on elliptic decompositions of the Guillemin-Kazhdan operators $η_\pm$ (Guillemin and Kazhdan, 1980) and leverages in the current setting the 0-calculus of Mazzeo-Melrose (Mazzeo and Melrose, 1987, Mazzeo, 1991). Iterating this decomposition gives rise to an "iterated-tt" representative modulo $\ker I_m$ for a tensor field, which is distinct from the often-used solenoidal representative. In the case of the Poincaré disk, we show that the X-ray transform of a tensor in iterated-tt form splits into components that are orthogonal relative to a specific $L^2$ structure in data space. We then provide a full picture of the data space decomposition, in particular a range characterization of $I_{m}$ for every $m$ in terms of moment conditions and spectral decay. Finally, we give explicit approaches for the reconstruction of tensors in iterated-tt form from their X-ray transform or its normal operator, using specific knowledge of geodesically invariant distributions with one-sided Fourier content, whose properties are analyzed in detail.

math.DG↗

arXiv2026

Gaussian-weighted normal operators on Euclidean space

We consider the normal operator of the X-ray transform, weighted with Gaussian weights, in Euclidean space with dimension at least 3. We show the eigenfunctions of the normal operator are joint eigenfunctions of the harmonic oscillator and the spherical Laplacian, and we relate the spectrum to that of elliptic operators in the 1-cusp pseudodifferential calculus.

math.AP↗

arXiv2026

A Comparison Test for Meromorphic Extensions

We provide a comparison test for meromorphic extensions, i.e., if two series are ``close enough" then the existence of a meromorphic extension of one to the entire complex plane ensures a similar extension for the other. We use this result to generate new examples of Dirichlet series admitting meromorphic extensions. Moreover, we demonstrate that our requirements are optimal by constructing a collection of counterexamples where the series are ``close but not enough": one series admits a meromorphic extension while the other possesses a natural boundary.

math.CV↗

arXiv2025

A Gutzwiller trace formula for singular potentials

The Gutzwiller trace formula relates the asymptotic spacing of quantum-mechanical energy levels in the semiclassical limit to the dynamics of periodic classical particle trajectories. We generalize this result to the case of non-smooth potentials, for which there is partial reflection of energy from derivative discontinuities of the potential. It is the periodic trajectories of an associated branching dynamics that contribute to the trace asymptotics in this more general setting; we obtain a precise description of their contribution.

math.AP↗