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Xiaolong Han

Publications and source records attributed to Xiaolong Han.

25 records · Page 2Linked to original sources
arXiv2016

Distribution of the nodal sets of eigenfunctions on analytic manifolds

The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on the manifold is analytic. We prove that if the eigenfunctions are equidistributed at a small scale, then the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold. In particular, on the negatively curved manifolds with analytic metric and on the tori, we show that in any eigenbasis, there is a full density subsequence of eigenfunctions such that the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold.

math.AP↗
arXiv2015

Small scale equidistribution of random eigenbases

We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity of eigenfrequency tends to infinity at least logarithmically. We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of multiplicity. In particular, this implies that almost surely random eigenbases on the n-dimensional sphere (n>=2) and the n-dimensional tori (n>=5) are equidistributed at polynomial scales.

math.SP↗
arXiv2015

Completeness of boundary traces of eigenfunctions

In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Bäcker, Fürstburger, Schubert, and Steiner, expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), is proved both for Dirichlet and Neumann boundary conditions. We then prove a semiclassical generalization of this identity.

math.AP↗
arXiv2014

Sharp norm estimates of layer potentials and operators at high frequency

In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency $λ^2\to\infty$. For single layer potentials, we find that the $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ norms decay in $λ$. The rate of decay depends on the curvature of $\partialΩ$: The norm is $λ^{-3/4}$ in general domains and $λ^{-5/6}$ if the boundary $\partialΩ$ is curved. The double layer potential, however, displays uniform $L^{2}(\partialΩ)\to{}L^{2}(Ω)$ bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. The appendix by Galkowski gives bounds $L^{2}(\partialΩ)\to{}L^{2}(\partialΩ)$ for the single and double layer operators at high frequency that are sharp modulo $\log λ$. In this case, both the single and double layer operator bounds depend upon the curvature of the boundary.

math.AP↗
arXiv2014

Spherical harmonics with maximal Lp (2<p<=6) norm growth

In this paper, we show that there exists a positive density subsequence of orthonormal spherical harmonics which achieves the maximal Lp norm growth for 2<p<=6, therefore giving an example of a Riemannian surface supporting such subsequence of eigenfunctions. This answers the question proposed by Sogge and Zelditch (arXiv:1011.0215). Furthermore, we provide an explicit lower bound on the density in this example.

math.CA↗
arXiv2013

Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group

In this paper, we investigate the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. On one hand, we apply the concentration compactness principle to prove the existence of the maximizers. While the approach here gives a different proof under the special cases discussed in a recent work of Frank and Lieb, we generalize the result to all admissible cases. On the other hand, we provide the upper bounds of sharp constants for these inequalities.

math.CA↗