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Xiaoqi Huang

Publications and source records attributed to Xiaoqi Huang.

28 records · Page 2Linked to original sources
arXiv2022

Global uniform in $N$ estimates for solutions of a system of Hartree-Fock-Bogoliubov type in the Gross-Pitaveskii regime

We extend the recent work of Chong et al., (2022) to the critical case. More precisely, we prove global in time, uniform in $N$ estimates for the solutions $ϕ$, $Λ$ and $Γ$ of a coupled system of Hartree--Fock--Bogoliubov type with interaction potential $\frac1NV_N(x-y)=N^{2}v(N(x-y))$. We assume that the potential $v$ is small which satisfies some technical conditions, and the initial conditions have finite energy. The main ingredient is a sharp estimate for the linear Schrödinger equation with potential in 6+1 dimension, which may be of interest in its own right.

math.AP↗
arXiv2022

Heat kernel estimate in a conical singular space

Let $(X,g)$ be a product cone with the metric $g=dr^2+r^2h$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the upper boundedness of heat kernel associated with the operator $L_V=-Δ_g+V_0 r^{-2}$, where $-Δ_g$ is the positive Friedrichs extension Laplacian on $X$ and $V=V_0(y) r^{-2}$ and $V_0\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $-Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$.The new ingredient of the proof is the Hadamard parametrix and finite propagation speed of wave operator on $Y$.

math.AP↗
arXiv2022

Product Manifolds with Improved Spectral Cluster and Weyl Remainder Estimates

We show that if $Y$ is a compact Riemannian manifold with improved $L^q$ eigenfunction estimates then, at least for large enough exponents, one always obtains improved $L^q$ bounds on the product manifold $X\times Y$ if $X$ is another compact manifold. Similarly, improved Weyl remainder term bounds on the spectral counting function of $Y$ lead to corresponding improvements on $X\times Y$. The latter results partly generalize recent ones of Iosevich and Wyman [14] involving products of spheres. Also, if $Y$ is a product of five or more spheres, we are able to obtain optimal $L^q(Y)$ and $L^q(X\times Y)$ eigenfunction and spectral cluster estimates for large $q$, which partly addresses a conjecture from [14] and is related to (and is partly based on) classical bounds for the number of integer lattice point on $λ\cdot S^{n-1}$ for $n\ge5$.

math.AP↗
arXiv2021

Uniform Sobolev Estimates on compact manifolds involving singular potentials

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author \cite{KRS} for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo \cite{DKS} for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the the first, third and fourth authors \cite{BSS} , Hassell and Tacy \cite{HassellTacy}, the first and fourth author \cite{SBLog}, and Hickman \cite{Hickman} as well as analogues of the improved uniform Sobolev estimates of \cite{BSSY} and \cite{Hickman} involving such potentials. Additionally, on $S^n$, we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author \cite{SHSo}. For general Riemannian manifolds we improve the earlier results in \cite{BSS} by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\in L^{n/2}$.

math.AP↗
arXiv2021

Quasimode and Strichartz estimates for time-dependent Schrödinger equations with singular potentials

We generalize the Strichartz estimates for Schrödinger operators on compact manifolds of Burq, Gérard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(I\times M)$-Strichartz estimates hold for $e^{-itH_V}$ when $H_V=-Δ_g+V(x)$ with $V\in L^{n/2}(M)$ if $n\ge3$ or $V\in L^{1+δ}(M)$, $δ>0$, if $n=2$, with $(p,q)$ being as in the Keel-Tao theorem and $I\subset {\mathbb R}$ a bounded interval. We do this by formulating and proving new "quasimode" estimates for scaled dyadic unperturbed Schrödinger operators and taking advantage of the the fact that $1/q'-1/q=2/n$ for the endpoint Strichartz estimates when $(p,q)=(2,2n/(n-2))$. We also show that the universal quasimode estimates that we obtain are saturated on {\em any} compact manifolds; however, we suggest that they may lend themselves to improved Strichartz estimates in certain geometries using recently developed "Kakeya-Nikodym" techniques developed to obtain improved eigenfunction estimates assuming, say, negative curvatures.

math.AP↗
arXiv2021

Weyl formulae for Schrödinger operators with critically singular potentials

We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators $H_V=-Δ_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials $V$. In particular, we extend the classical results of Avakumović , Levitan and Hörmander by obtaining $O(λ^{n-1})$ bounds for the error term in the Weyl formula in the universal case when we merely assume that $V$ belongs to the Kato class, ${\mathcal K}(M)$, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. In this case, we can also obtain extensions of the Duistermaat-Guillemin theorem yielding $o(λ^{n-1})$ bounds for the error term under generic conditions on the geodesic flow, and we can also extend Bérard's theorem yielding $O(λ^{n-1}/\log λ)$ error bounds under the assumption that the principal curvatures are non-positive everywhere. We can obtain further improvements for tori, which are essentially optimal, if we strengthen the assumption on the potential to $V\in L^p(M)\cap {\mathcal K}(M)$ for appropriate exponents $p=p_n$.

math.AP↗
arXiv2021

Uniform Sobolev estimates in $\mathbb{R}^{n}$ involving singular potentials

We generalize the Stein-Tomas [17] $L^2$-restricition theorem and the uniform Sobolev estimates of Kenig, Ruiz and the second author [11] by allowing critically singular potential. We also obtain Strichartz estimates for Schrödinger and wave operators with such potentials. Due to the fact that there may be nontrivial eigenfunctions we are required to make certain spectral assumptions, such as assuming that the solutions only involve sufficiently large frequencies.

math.AP↗
arXiv2020

$L^p$ eigenfunction bounds for fractional Schrödinger operators on manifolds

This paper is dedicated to $L^p$ bounds on eigenfunctions of a Schödinger-type operator $(-Δ_g)^{α/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-Δ_g)^{α/2}$ is defined spectrally in terms of the eigenfunctions of $-Δ_g$. We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation $(\partial_t^2+(-Δ_g)^{α/2}+V)u=0$. The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao and Shao-Yao play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the "uniform Sobolev estimates" by Kenig-Ruiz-Sogge. These enable us to handle the critically singular potentials $V$ and prove the quasimode estimates.

math.AP↗
arXiv2019

Restriction of toral eigenfunctions to totally geodesic submanifolds

We estimate the $L^2$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus $\mathbb{T}^d$, $d\ge2$. We reduce getting correct bounds to counting lattice points in the intersection of some $ν$-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain-Rudnick on $L^2$-restriction estimates for rational hyperplanes. On $\mathbb{T}^2$, we prove the uniform $L^2$ restriction bounds for closed geodesics. On $\mathbb{T}^3$, we obtain explicit $L^2$ restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq-Gérard-Tzvetkov, Hu, Chen-Sogge.

math.AP↗