arXiv subjects

Qiuye Jia

Publications and source records attributed to Qiuye Jia.

At least 19 recordsLinked to original sources

arXiv2026

Sharp Dispersive Estimates for the Schrödinger Equation with an Attractive Coulomb Potential

We prove sharp dispersive $L^1 \to L^\infty$ estimates for the three-dimensional attractive Coulomb operator $H_Z=-Δ-Z|x|^{-1}$, where $Z>0$. The absolutely continuous part of the Schrödinger evolution decays at the free rate for short times, whereas its leading contribution decays like $|t|^{-1}$ for long times, with amplitude proportional to $Z$. This slower decay is driven by the threshold and is sharp when $Z^2|t|\gg1$.

math.AP↗

arXiv2026

Scattering for the focusing $H^{1/2}$-critical nonlinear Schrödinger equation with large data

In this article we prove that the solution to the focusing $H^{1/2}$-critical nonlinear Schrödinger equation in dimension $d\geq 5$ scatters outside finitely many arbitrarily small spacetime cones. We will discuss the relationship between Theorem~\ref{thm:finite-bad-directions} and the resolution of solitons. There are two key ingredients in the proof: the localized below-ground-state scattering in Theorem~\ref{thm:cone-scattering}, and the characterization in Theorem~\ref{thm:measure-convergence} of the asymptotic behaviour of the residual part (i.e., after subtracting the scattering part). As a byproduct, we also establish an upgrading machinery: if the solution is below the ground state (resp. small) along a time sequence in a spacetime cone, then it is also below ground state (resp. small) uniformly for large time in a shrinked spacetime cone. We expect this to be useful in future works using concentration compactness and our spacetime cone approach.

math.AP↗

arXiv2026

The effect of geometric focusing on dispersive estimates for the Schrödinger and wave equations

We classify the long-time decay rate in dispersive estimates for the Schrödinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones in terms of the intensity of geometric focusing. Letting $X_0$ be a metric cone, one of our main results demonstrates that each multiplicity of conjugate points within distance $π$ on $Y=\partial X_0$ leads to a $|t|^{1/2}$-loss in the long-time decay order and a half-order shift in the regularity index in the dispersive estimate for the Schrödinger equation. Unexpectedly, conjugate point pairs on $Y$ at distance $π$ do not cause loss when the Legendre submanifold carrying the wave propagation satisfies a natural admissible condition that we propose. In sum, we give a robust framework for proving dispersive estimates that is stable under geometric perturbations and also accommodates perturbations by potentials.

math.AP↗

arXiv2026

Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space

We study the pointwise decay estimates for the Schrödinger and wave equations on a product cone $(X,g)$, where the metric $g=dr^2+r^2 h$ and $X=C(Y)=(0,\infty)\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. Under the assumption that the {conjugate radius} $\conR$ of $Y$ satisfies $\conR>π$, we prove the pointwise dispersive estimates for the Schrödinger and half-wave propagators in this setting. The key ingredient is the modified Hadamard parametrix on $Y$ in which the role of the conjugate points does not come into play if $\conR>π$. A new finding is that a threshold of the {conjugate radius} of $Y$ for the pointwise dispersive estimates in this setting is the magical number $π$.

math.AP↗

arXiv2026

The topological pressure of trapped sets in Kerr-(de Sitter) spacetimes

In this paper we prove that the topological pressure of dynamic systems with normally hyperbolic trapping is negative. In particular, this applies to the null geodesic flow in Kerr and Kerr-de Sitter spacetimes This builds connection between results for trapped sets with low regularity in hyperbolic dynamic systems conditioning on negativity of the topological pressure and unconditional results in the setting of normally hyperbolic trapping.

math.DS↗

arXiv2026

Determining potentials from the scattering map of the time-dependent Schrödinger equation

For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of a solution as $t \to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for a certain class of metrics, the scattering maps and Poisson operators associated to two Schrödinger operators on the same curved space only differ by a compact operator on a critical level if and only if the two potentials are equal.

math.AP↗

arXiv2026

Determining metrics from the scattering map of the time-dependent Schrödinger equation

For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of solution as $t\to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for certain class of metrics, the scattering maps associated to two Schrödinger operators with two time dependent metrics only differ by a compact operator if and only if these two metrics are related by a pull-back of a diffeomorphism.

math.AP↗

arXiv2026

Lecture notes on non-elliptic Fredholm theory

These are lecture notes from the Austral Winter School on Microlocal Analysis and Non-elliptic Fredholm Theory, held at the Australian National University, Canberra, June 30 -- July 11, 2025.

math.AP↗

arXiv2026

The scattering map for the Schrodinger operator on curved spaces

Let $P$ be a Schrödinger operator $D_t+Δ_g$ with metric and potential perturbation that are compactly supported in spacetime $\mathbb{R}^{n+1}$. Here $D_t = -i \partial_t$ and $Δ_g$ is the positive Laplacian. We consider the scattering map $S$ defined previously by the first author with Gell-Redman and Gomes arXiv:2201.03140, which relates the asymptotic data, as $t \to \pm \infty$, of global solutions $u$ to $Pu = 0$. We show that $S$ is a `1-cusp' Fourier integral operator, where `1-cusp' refers to a pseudodifferential calculus introduced by Vasy and Zachos arXiv:2204.11706 in the completely different setting of inverse problems on asymptotically conic manifolds. Our viewpoint is that 1-cusp geometry is the natural setting for studying the asymptotic data of solutions to Schrödinger's equation.

math.AP↗

arXiv2025

Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics

This is the less technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrödinger equation. As shown in the companion paper, combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. In this paper, we derive asymptotics from those estimates. Our framework differs from those in previous works in that ours is based on spacetime phase-space analysis.

math.AP↗

arXiv2025

Bochner-Riesz means on a conical singular manifold

We prove a sharp $L^p$-boundedness criterion for Bochner-Riesz multipliers on flat cones $X = (0,\infty) \times \mathbb{S}_σ^1$. The operator $S_λ^δ(Δ_X)$ is bounded on $L^p(X)$ for $1 \leq p \leq \infty$, $p \neq 2$, if and only if $δ> δ_c(p,2) = \max\left\{ 0, 2\left| 1/2 - 1/p \right| - 1/2 \right\}$. This result is also applicable to the infinite sector domain with Dirichlet or Neumann boundary, resolving the critical exponent problem in this wedge setting.

math.AP↗

arXiv2025

The essential self-adjointness of the wave operator on radiative spacetimes

We prove the essential self-adjointness of the d'Alembertian $\square_g$, allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in closely related settings, that all tempered distributions $u$ satisfying $\square_g u = λu +f$ for $λ\in \mathbb{C}\backslash \mathbb{R}$ and $f$ Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the ``de,sc-'' machinery recently developed by the third author.

math.SP↗

arXiv2025

Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates

This is the more technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrodinger equation. Combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. The companion paper gives applications. Our main technical tools are three new pseudodifferential calculi, $Ψ_{\natural}$ (a variant of the semiclassical scattering calculus), $Ψ_{\natural\mathrm{res}}$, and $Ψ_{\natural2\mathrm{res}}$, the latter two of which are created by ``second microlocalizing'' the first at certain locations. This paper and the companion paper can be read in either order, since the latter treats the former as a black box.

math.AP↗

arXiv2025

Decay and Strichartz estimates for critical electromagnetic wave equations on conic manifolds

We establish the decay and Strichartz estimates for the wave equation with large scaling-critical electromagnetic potentials on a conical singular space $(X,g)$ with dimension $n\geq3$, where the metric $g=dr^2+r^2 h$ and $X=C(Y)=(0,\infty)\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. The decay assumption on the magnetic potentials is scaling critical and includes the decay of Coulomb type. The main technical innovation lies in proving localized pointwise estimates for the half-wave propagator by constructing a localized spectral measure, which effectively separates contributions from conjugate point pairs on $\CS$. In particular, when $Y=\mathbb{S}^{n-1}$, our results, which address the case of large critical electromagnetic potentials, extend and improve upon those in [21], which considered sufficiently decaying, and small potentials and that of [24], which considered potentials decaying faster than scaling critical ones.

math.AP↗

arXiv2025

Strichartz estimates for the Schrödinger equation in high dimensional critical electromagnetic fields

We prove Strichartz estimates for the Schrödinger equation with scaling-critical electromagnetic potentials in dimensions $n\geq3$. The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential. Our approach introduces novel techniques, notably the construction of Schwartz kernels for the localized Schrödinger propagator, which separates the antipodal points of $\mathbb{S}^{n-1}$, in these scaling critical electromagnetic fields. This method enables us to prove the $L^1(\mathbb{R}^n)\to L^\infty(\mathbb{R}^n)$ for the localized Schrödinger propagator, as well as global Strichartz estimates. Our results provide a positive answer to the open problem posed in arXiv:0901.4024 arXiv:1611.04805 arXiv:0806.0778, and fill a longstanding gap left by arXiv:arXiv:0705.0546 arXiv:archive/0608699.

math.AP↗

arXiv2024

2D inverse problem with a Foliation Condition

The inverse problem of X-tray transforms considers reconstructing functions from some data that are easier to measure, which is typically the integral of that function along geodesics. We prove that if the domain has a foliation structure, then this reconstruction process is possible on certain function classes.

math.AP↗

arXiv2024

Propagation of Singularity with Normally Hyperbolic Trapping

We prove microlocal estimates with normally hyperbolic trapping. We use a new type of symbol class which is constructed by blowing up the intersection of the unstable manifold and the fiber infinity. For scalar wave equations on Kerr(-de Sitter) spacetimes, the extra loss of the microlocal estimates compared with the standard propagation of singularities is arbitrarily small.

math.AP↗

arXiv2024

The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3

In this article we consider the defocusing nonlinear Schrödinger equation, with time-dependent potential, in space dimensions $n=1, 2$ and $3$, with nonlinearity $|u|^{p-1} u$, $p$ an odd integer, satisfying $p \geq 5$ in dimension $1$, $p \geq 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k \geq 2$ under the action of certain vector fields generating symmetries of the free Schrödinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state.

math.AP↗