arXiv subjects

Xiaofen Gao

Publications and source records attributed to Xiaofen Gao.

6 recordsLinked to original sources

arXiv2025

Strichartz estimates for Critical magnetic Schrödinger operators on flat Euclidean cones

In this paper, we study Schrödinger operator $\mathcal{H}_{\mathbf{A}}$ perturbed by critical magnetic potentials on the 2D flat cone $Σ= C(\mathbb{S}_ρ^1) = (0, \infty) \times \mathbb{S}_ρ^1$, which is a product cone over the circle $\mathbb{S}_ρ^1 = \mathbb{R}/2πρ\mathbb{Z}$ with radius $ρ> 0$, and equipped with the metric $g = dr^2 + r^2 dθ^2$. The goal of this work is to establish Strichartz estimates for $\mathcal{H}_{\mathbf{A}}$ in this setting. A key aspect of our approach is the construction of the Schwartz kernel of the resolvent and the spectral measure for Schrödinger operator on the flat Euclidean cone $(Σ, g)$. The results presented here generalize previous work in \cite{Ford, BFM, FZZ, Zhang1}.

math.AP↗

arXiv2025

$L^p$-estimates for the wave equation with critical magnetic potential on conical manifolds

In this paper, we consider a class of conical singular spaces $Σ=(0,\infty)_r\times Y$ equipped with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$ without boundary. In this context, we aim to show that the sine wave propagator $\sin\left(t\sqrt{\mathcal{L}_{\mathbf{A}}}\right)/\sqrt{\mathcal{L}_{\mathbf{A}}}$ is bounded in $L^{p}(Σ)$, where $\mathcal{L}_{\mathbf{A}}$ is a magnetic Schrödinger operator with a scaling-critical magnetic potential on metric cone $Σ$. Our main result is the generalization of the result in \cite{L}. The novel ingredient is the construction of Hadamard parametrix for $\cos\left(t\sqrt{\mathcal{L}_{\bf A}}\right)$ on $Σ$.

math.AP↗

arXiv2021

Dispersive estimates for two-particles Schrödinger and wave equations in the Aharonov-Bohm field

We study the dispersive behaviors of two-particles Schrödinger and wave equations in the Aharonov-Bohm field. In particular, we prove the Strichartz estimates for Schrödinger and wave equations in this setting. The key point is to construct spectral measure of Schrödinger operator with an Aharonov-Bohm type potential in $\R^4$. As applications, we finally prove a scattering theory for the nonlinear defocusing subcritical two-particles Schrödinger equation with Aharonov-Bohm potential.

math.AP↗

arXiv2021

Decay and Strichartz estimates in critical electromagnetic fields

We study the $L^1\to L^\infty$-decay estimates for dispersive equations in the Aharonov-Bohm magnetic fields, and further prove Strichartz estimates for the Klein-Gordon equation with critical electromagnetic potentials. The novel ingredients are the construction of Schwartz kernels of the spectral measure and heat propagator for the Schrödinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schrödinger operator with Aharonov-Bohm potentials, and show that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. In future papers, this result on the spectral measure will be used to (i) study the uniform resolvent estimates, and (ii) prove the $L^p$-regularity property of wave propagation in the same setting.

math.AP↗

arXiv2020

Restriction estimates in a conical singular space: wave equation

We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. Let $Δ_g$ be the Friedrich extension positive Laplacian on $X$, and consider the operator $\mathcal{L}_V=Δ_g+V$ with $V=V_0r^{-2}$, where $V_0(θ)\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $Δ_h+V_0+(n-2)^2/4$ is positive. In the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with $\mathcal{L}_V$. The smallest positive eigenvalue of the operator $Δ_h+V_0+(n-2)^2/4$ plays an important role in the result. As an application, for independent of interests, we prove local energy estimates and Keel-Smith-Sogge estimates for the wave equation in this setting.

math.AP↗

arXiv2019

Scattering theory for nls with inverse-square potential in 2d

In this paper, we study the long time behavior of the solution of nonlinear Schrödinger equation with a singular potential. We prove scattering below the ground state for the radial NLS with inverse-square potential in dimension two $$iu_t+Δu-\frac{a u}{|x|^2}= -|u|^pu$$ when $2 0$. This work extends the result in [13, 14, 16] to dimension 2D. The key point is a modified version of Arora-Dodson-Murphy's approach [2].

math.AP↗