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Xue Ping Wang

Publications and source records attributed to Xue Ping Wang.

11 recordsLinked to original sources
arXiv2025

The Kramers-Fokker-Planck equation with a decaying potential in $\mathbb R^n$, $n \ge 4$

We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in $\mathbb R^n$, $n \ge 4$, and deduce the large-time behavior of solutions to the kinetic Kramers-Fokker-Planck equation. For short-range potentials, we establish an optimal time-decay estimate in weighted $L^2$-spaces when $ n\ge 5$ is odd. For potentials decaying like $O(|x|^{-ρ})$ for some $ρ> n-1$, we obtain, for all dimensions $n \ge 4$, a large-time expansion of the solution with the leading term given by the Maxwell-Boltzmann distribution multiplied by the factor $(4πt)^{-\frac n 2}$ corresponding to the decay for the heat equation. These results complete those obtained in [16, 22] for dimensions $n=1$ and $3$. The same questions for $n=2$ are still open.

math.AP↗
arXiv2024

Spectral properties of the Kramers-Fokker-Planck operator with a long-range potential

We study real resonances and embedded eigenvalues of the Kramers--Fokker--Planck operator with a long-range potential. We prove that thresholds are only possible accumulation points of eigenvalues and that the limiting absorption principle holds true for energies outside an exceptional set. We also prove that the eigenfunctions associated with discrete eigenvalues decay exponentially and those associated with embedded non-threshold ones decay polynomially.

math.AP↗
arXiv2023

Spectral analysis of $N$-body Schrödinger operators at two-cluster thresholds

This book provides a systematic study of spectral and scattering theory for many-body Schrödinger operators at two-cluster thresholds. While the two-body problem (reduced after separation of the center of mass motion to a one-body problem at zero energy) is a well-studied subject, the literature on the many-body problem is sparse. However our analysis covers for example the system of three particles interacting by Coulomb potentials and restricted to a small energy region to the right of a fixed nonzero two-body eigenvalue $λ_0$. In general we address the question: How does scattering quantities for the many-body atomic and molecular models behave in the limit when the total energy approaches a fixed two-cluster threshold $λ_0$? This includes mapping properties and singularities of the limiting scattering matrix, asymptotics of the total scattering cross-section and absence of transmission from one channel to another in the small inter-cluster kinetic energy region. Our principal tools are the Feshbach-Grushin dimension reduction method and spectral analysis based on a certain Mourre estimate. Additional topics (of independent interest) are the limiting absorption principle, micro-local resolvent estimates, Rellich and Sommerfeld type theorems and asymptotics of the limiting resolvents at thresholds. While all of these features are fairly well-understood for two-body Schrödinger operators, they are poorly understood in the many-body case even for two-cluster thresholds. It is the goal of the book to remedy this point. The mathematical physics field under study is very rich, and there are many open problems, several of them stated explicitly in the book for the interested reader.

math-ph↗
arXiv2021

Global-in-time L p -- L q estimates for solutions of the Kramers-Fokker-Planck equation

In this work, we prove an optimal global-in-time L p --L q estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as t $\rightarrow$ +$\infty$ is the same as the heat equation in x-variables and the divergence rate as t $\rightarrow$ 0 + is related to the sub-ellipticity with loss of 1/3 derivatives of the Kramers-Fokker-Planck operator.

math.AP↗
arXiv2017

Gevrey estimates of the resolvent and sub-exponential time-decay of solutions

In this article, we study a class of non-selfadjoint Schr{ö}dinger operators H which are perturbation of some model operator H 0 satisfying a weighted coercive assumption. For the model operator H 0 , we prove that the derivatives of the resolvent satisfy some Gevrey estimates at threshold zero. As application, we establish large time expansions of semigroups e --tH and e --itH for t > 0 with subexponential time-decay estimates on the remainder, including possible presence of zero eigenvalue and real resonances.

math.AP↗
arXiv2014

Large-time asymptotics of solutions to the Kramers-Fokker-Planck equation with a short-range potential

In this work, we use scattering method to study the Kramers-Fokker-Planck equation with a potential whose gradient tends to zero at the infinity. For short-range potentials in dimension three, we show that complex eigenvalues do not accumulate at low-energies and establish the low-energy resolvent asymptotics. This combined with high energy pseudospectral estimates valid in more general situations gives the large-time asymptotics of the solution in weighted $L^2$ spaces.

math.SP↗
arXiv2010

A new Levinson's theorem for potentials with critical decay

We study the low-energy asymptotics of the spectral shift function for Schrödinger operators with potentials decaying like $O(\frac{1}{|x|^2})$. We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue and zero resonance.

math.SP↗
arXiv2010

Two-body threshold spectral analysis, the critical case

We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schrödinger operators with a radially symmetric potential falling off like $-γr^{-2},\;γ>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $γ>(l+d/2-1)^2$. In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.

math.SP↗
arXiv2009

Number of eigenvalues for a class of non-selfadjoint Schrödinger operators

In this article, we prove the finiteness of the number of eigenvalues for a class of Schrödinger operators $H = -Δ+ V(x)$ with a complex-valued potential $V(x)$ on $\bR^n$, $n \ge 2$. If $\Im V$ is sufficiently small, $\Im V \le 0$ and $\Im V \neq 0$, we show that $N(V) = N(\Re V)+ k$, where $k$ is the multiplicity of the zero resonance of the selfadjoint operator $-Δ+ \Re V$ and $N(W)$ the number of eigenvalues of $-Δ+ W$, counted according to their algebraic multiplicity.

math.SP↗