arXiv subjects

Junyong Zhang

Publications and source records attributed to Junyong Zhang.

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

Dispersive estimates for the Landau Hamiltonian on the hyperbolic plane

In this paper, We obtain dispersive estimates for solutions to the Schrödinger equation with a uniform magnetic field on the hyperbolic plane $\mathbb{H}$. The key ingredient is an explicit representation formula for the kernel of the corresponding Schrödinger propagator. As a consequence, we prove the corresponding Strichartz estimates for all admissible pairs on $\mathbb{H}$.

math.AP↗

arXiv2026

$L^p$ bounds for wave operators with critical electromagnetic potentials

We study the Møller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside $\frac12\Z$, we prove that the wave operators relative to $-Δ$ exist, are unitary on $L^2$, and, together with their adjoints, are bounded on every $L^p$, $1<p<\infty$. We then specialize to the the Aharonov--Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted $L^p(\R^2,|x|^β\,dx)$ spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all $L^{p}$ spaces $1<p<\infty$, while boundedness fails at $p=1,\infty$. In the Krein case, both wave operators and adjoints are bounded precisely when $2/(2-η_α)<p<2/η_α$, where $η_α=\max\{α,1-α\}$ (here $α\in(0,1)$). Thus the boundary condition changes the admissible exponents.

math.SP↗

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

Strichartz estimates for Schrödinger equations with the multipole Aharonov--Bohm Hamiltonian

We prove global-in-time Strichartz estimates for Schrödinger equations with multipole Aharonov--Bohm Hamiltonians on $\mathbb{R}^2$. As intermediate steps, we prove global-in-time local smoothing estimates for multipole Aharonov--Bohm Hamiltonians.

math.AP↗

arXiv2026

$L^p$-estimates for the wave equation with partial inverse-square potentials

This paper investigates $L^p$-estimates for solutions to the wave equation perturbed by a scaling-critical partial inverse-square potential. We study a model in which the singularity of the potential appears only in a subset of the variables, corresponding to the Schrödinger operator $\mathcal{H}_a = -Δ_x - Δ_y + a/|x|^2$ on $\mathbb{R}^{2+n}$. Using spectral analysis, we establish the $L^p$-boundedness of the wave propagator $(1+\sqrt{\mathcal{H}_a})^{-γ} e^{it\sqrt{\mathcal{H}_a}}$ for a range of exponents $γ$ and $p$ satisfying $|1/p -1/2| < γ/(n+1)$. The key ingredients are the spectral measure kernel of the partial inverse-square operator $\mathcal{H}_a$ and the complex interpolation argument.

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

Negative Order Bochner-Riesz Operators for the Critical Magnetic Schrödinger Operator in $\mathbb{R}^2$

This paper studies the sharp $L^p$-$L^q$ boundedness of the Bochner-Riesz operator $S^δ_λ(\mathcal{L}_{\mathbf{A}})$ associated with a scaling-critical magnetic Schrödinger operator $\mathcal{L}_{\mathbf{A}}$ on $\mathbb{R}^2$, where $δ\in (-3/2, 0)$. We determine the conditions on the exponents $p$ and $q$ under which the operator is bounded from $L^p(\mathbb{R}^2)$ to $L^q(\mathbb{R}^2)$. Our main result characterizes the boundedness region as a pentagonal subset $Δ(δ)$ of the $(1/p, 1/q)$-plane, extending previous uniform resolvent result in Fanelli, Zhang and Zheng[Int. Math. Res. Not., 20(2023), 17656-17703].

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↗

arXiv2025

Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case

In this paper we study the dispersive properties of a two dimensional massless Dirac equation perturbed by an Aharonov--Bohm magnetic field. Our main results will be a family of pointwise decay estimates and a full range family Strichartz estimates for the flow. The proof relies on the use of a relativistic Hankel transform, which allows for an explicit representation of the propagator in terms of the generalized eigenfunctions of the operator. These results represent the natural continuation of earlier research on evolution equations associated to operators with magnetic fields with strong singularities (see \cite{DF, FFFP, FZZ} where the Schrödinger and the wave equations were studied). Indeed, we recall the fact that the Aharonov--Bohm field represents a perturbation which is critical with respect to the scaling: this fact, as it is well known, makes the analysis particularly challenging.

math.AP↗

arXiv2025

$L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field

In this paper, we study the $L^{p}$-estimates for the solution to the $2\mathrm{D}$-wave equation with a scaling-critical magnetic potential. Inspired by the work of \cite{FZZ}, we show that the operators $(I+\mathcal{L}_{\mathbf{A}})^{-γ}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}$ is bounded in $L^{p}(\mathbb{R}^{2})$ for $1 |1/p-1/2|$ and $t>0$, where $\mathcal{L}_{\mathbf{A}}$ is a magnetic Schrödinger operator. In particular, we derive the $L^{p}$-bounds for the sine wave propagator $\sin(t\sqrt{\mathcal{L}_{\mathbf{A}}})\mathcal{L}^{-\frac12}_{\mathbf{A}}$. The key ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.

math.AP↗

arXiv2025

Dispersive and Strichartz estimates for Dirac equation in a cosmic string spacetime

In this work we study the Dirac equation on the cosmic string background, which models a one--dimensional topological defect in the spacetime. We first define the Dirac operator in this setting, classifying all of its selfadjoint extensions, and we give an explicit kernel for the propagator. Secondly, we prove dispersive estimates for the flow, with and without weights. Finally, we prove Strichartz estimates for the flow in a sharp restricted set of indices, which are different from the classical Euclidean ones.

math.AP↗

arXiv2024

Intertwining operators beyond the Stark Effect

The main mathematical manifestation of the Stark effect in quantum mechanics is the shift and the formation of clusters of eigenvalues when a spherical Hamiltonian is perturbed by lower order terms. Understanding this mechanism turned out to be fundamental in the description of the large-time asymptotics of the associated Schrödinger groups and can be responsible for the lack of dispersion in Fanelli, Felli, Fontelos and Primo [Comm. Math. Phys., 324(2013), 1033-1067; 337(2015), 1515-1533]. Recently, Miao, Su, and Zheng introduced in [Tran. Amer. Math. Soc., 376(2023), 1739--1797] a family of spectrally projected intertwining operators, reminiscent of the Kato's wave operators, in the case of constant perturbations on the sphere (inverse-square potential), and also proved their boundedness in $L^p$. Our aim is to establish a general framework in which some suitable intertwining operators can be defined also for non constant spherical perturbations in space dimensions 2 and higher. In addition, we investigate the mapping properties between $L^p$-spaces of these operators. In 2D, we prove a complete result, for the Schrödinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and $L^p$-bounds of Bochner--Riesz means. In higher dimensions, apart from recovering the example of inverse-square potential, we can conjecture a complete result in presence of some symmetries (zonal potentials), and open some interesting spectral problems concerning the asymptotics of eigenfunctions.

math.AP↗

arXiv2024

Massless Dirac equation on spinor bundles over real hyperbolic spaces

We prove a sharp-in-time dispersive estimate of the Dirac equation on spinor bundles over the real hyperbolic space. Compared with the Euclidean counterparts, our result shows that the dispersive estimate differs between short and long times, reflecting the intuitive influence of negative curvature on the dispersion. Moreover, the well-known equivalence between dispersive estimates for Dirac and wave propagators in the Euclidean setting no longer holds in this context. This finding suggests that spinor fields are affected by the geometry at infinity of the manifold. As a key application, we establish an improved global-in-time Strichartz estimate, in the sense that there is no loss of angular derivatives and the admissible set is larger than previously known results in other settings.

math.AP↗

arXiv2024

Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space

In this paper, we study Strichartz estimates for the Schrödinger equation on a metric cone $X$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. For the metric $g$ on $X$ given by $g=dr^2+r^2h$, let $Δ_g$ be the positive Friedrichs extension Laplacian on $X$ and $V=V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ is a real function such that the operator $P:=Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. We establish the full range of global-in-time Strichartz estimates without loss for the Schrödinger equation associated with the operator $\LL_V=Δ_g+V_0 r^{-2}$ including the endpoint estimate both in homogeneous and inhomogeneous cases. A new finding reveals that the range of admissible pairs at $\dot H^s$-level is influenced by the smallest eigenvalue of the operator $P$. This additionally proves the conjecture in Wang [Ann. Inst. Fourier 2006] and generalizes the results of Ford [Comm. Math. Phys. 2010] and Baskin-Marzuola-Wunsch [Contemp. Math. 2014]. As an application, we show the well-posedness theory and scattering theory for the Schrödinger equation with a cubic nonlinearity on this setting which verifies a conjecture in Baskin-Marzuola-Wunsch [Contemp. Math. 2014].

math.AP↗

arXiv2024

Bochner-Riesz means for critical magnetic Schrödinger operators in ${\mathbb R^2}$

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schrödinger operators $\mathcal{L}_{\bf A}$ in ${\mathbb{R}^2}$, which involve the physcial Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\neq 2$, the Bochner-Riesz operator $S_λ^δ(\mathcal{L}_{\bf A})$ of order $δ$ is bounded on $L^p(\mathbb{R}^2)$ if and only if $δ>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient of the proof is to obtain the localized $L^4(\mathbb R^2)$ estimate of $S_λ^δ(\mathcal{L}_{\bf A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_λ^δ(Δ)$ for the Laplacian $Δ$ in $\mathbb{R}^2$.

math.AP↗