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Yuzhou Zou

Publications and source records attributed to Yuzhou Zou.

9 recordsLinked to original sources
arXiv2026

The hyperbolic X-ray transform: new range characterizations, mapping properties and functional relations

We derive new singular value decompositions and range characterizations for the X-ray transform on the Poincaré disk, intertwining relations with distinguished differential operators of wedge type, and a surjectivity result for the backprojection operator. New functional settings are found, which allow to sharply understand boundary behavior issues and invertibility settings. The approach mainly exploits analogous results obtained only recently in the Euclidean disk, together with the projective equivalence between the two models.

math.AP↗
arXiv2024

Asymptotic Expansion of the Eigenvalues of a Bathtub Potential with Quadratic Ends

We consider the eigenvalues of a one-dimensional semiclassical Schrödinger operator, where the potential consist of two quadratic ends (that is, looks like a harmonic oscillator at each infinite end), possibly with a flat region in the middle. Such a potential notably has a discontinuity in the second derivative. We derive an asymptotic expansion, valid either in the high energy regime or the semiclassical regime, with a leading order term given by the Bohr-Sommerfeld quantization condition, and an asymptotic expansion consisting of negative powers of the leading order term, with coefficients that are oscillatory in the leading order term. We apply this expansion to study the results of the Gutzwiller Trace formula and the heat kernel asymptotic for this class of potentials, giving an idea into what results to expect for such trace formulas for non-smooth potentials.

math-ph↗
arXiv2024

Helmholtz quasi-resonances are unstable under most single-signed perturbations of the wave speed

We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter $z$. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We show that there exists a polynomial function of frequency such that, for any frequency, for most values of $z$, the norm of the solution operator is bounded by that function. This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows superalgebraically, with these frequencies often called $\textit{quasi-resonances}$. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations.

math.AP↗
arXiv2023

Boundary triples for a family of degenerate elliptic operators of Keldysh type

We consider a one-parameter family of degenerately elliptic operators $\cal{L}_γ$ on the closed disk $\mathbb{D}$, of Keldysh (or Kimura) type, which appears in prior work [Mishra et al., Inverse Problems (2022)] by the authors and Mishra, related to the geodesic X-ray transform. Depending on the value of a constant $γ\in \mathbb{R}$ in the sub-principal term, we prove that either the minimal operator is self-adjoint (case $|γ|\ge 1$), or that one may construct appropriate trace maps and Sobolev scales (on $\mathbb{D}$ and $\mathbb{S}^1=\partial\mathbb{D}$) on which to formulate mapping properties, Dirichlet-to-Neumann maps, and extend Green's identities (case $|γ|<1$). The latter can be reinterpreted in terms of a boundary triple for the maximal operator, or a generalized boundary triple for a distinguished restriction of it. The latter concepts, object of interest in their own right, provide avenues to describe sufficient conditions for self-adjointness of extensions of $\cal{L}_{γ,min}$ that are parameterized in terms of boundary relations, and we formulate some corollaries to that effect.

math.AP↗
arXiv2022

The $C^\infty$-isomorphism property for a class of singularly-weighted X-ray transforms

We study a one-parameter family of self-adjoint normal operators for the X-ray transform on the closed Euclidean disk ${\mathbb D}$, obtained by considering specific singularly weighted $L^2$ topologies. We first recover the well-known Singular Value Decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of $C^\infty({\mathbb D})$. As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the X-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior.

math.AP↗
arXiv2022

Partial Global Recovery in the Elastic Travel Time Tomography Problem for Transversely Isotropic Media

We consider the problem of recovering material parameters in a transversely isotropic medium from the qP and qSV waves' travel times, given the axis of isotropy and the material parameters associated to the qSH wave speed. The operators obtained from the pseudolinearization argument are of parabolic type, and so we discuss inverting operators whose symbols are of parabolic type. We present stability estimates for recovering either one parameter from one wave speed or two parameters from two wave speeds with the remaining parameters either known or with a known functional relationship, and these estimates provide injectivity among parameters that differ on sets of small width.

math.AP↗
arXiv2021

Microlocal Methods for The Elastic Travel Time Tomography Problem for Transversely Isotropic Media

We study the travel time tomography problem for transversely isotropic media using the artificial boundary method used in works in X-ray tomography and boundary rigidity problems. The analysis involved requires a modification of the scattering pseudodifferential calculus in order to provide parametrices for operators which are degenerately elliptic; such a calculus is constructed in this paper and used to solve the tomography problem of interest.

math.AP↗
arXiv2020

Streak artifacts from non-convex metal objects in X-ray tomography

We study artifacts in the reconstruction of X-ray tomography due to nonlinear effects. For non-convex metal objects, we analyze the new phenomena of streak artifacts from inflection points on the boundary of metal objects. We characterize the location and strength of all possible artifacts using notions of conormal distributions associated with the proper geometry.

math.AP↗