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Jingwei Guo

Publications and source records attributed to Jingwei Guo.

29 records · Page 2Linked to original sources
arXiv2019

The Weyl formula for planar annuli

We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in $\mathbb{R}^2$. Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amount and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size $O(μ^{2/3})$. If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely $O(μ^{131/208}(\log μ)^{18627/8320})$. As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

math.SP↗
arXiv2017

Lattice points in stretched model domains of finite type in $\mathbb{R}^d$

We study an optimal stretching problem for certain convex domain in $\mathbb{R}^d$ ($d\geq 3$) whose boundary has points of vanishing Gaussian curvature. We prove that the optimal domain which contains the most positive (or least nonnegative) lattice points is asymptotically balanced. This type of problem has its origin in the "eigenvalue optimization among rectangles" problem in spectral geometry. Our proof relies on two-term bounds for lattice counting for general convex domains in $\mathbb{R}^d$ and an explicit estimate of the Fourier transform of the characteristic function associated with the specific domain under consideration.

math.NT↗
arXiv2017

An improved remainder estimate in the Weyl formula for the planar disk

In \cite{colin}, Y. Colin de Verdière proved that the remainder term in the two-term Weyl formula for the eigenvalue counting function for the Dirichlet Laplacian associated with the planar disk is of order $O(λ^{2/3})$. In this paper, by combining with the method of exponential sum estimation, we will give a sharper remainder term estimate $O(λ^{2/3-1/495})$.

math.SP↗
arXiv2013

Lattice points in rotated convex domains

If $\mathcal{B}\subset \mathbb{R}^d$ ($d\geqslant 2$) is a compact convex domain with a smooth boundary of finite type, we prove that for almost every rotation $θ\in SO(d)$ the remainder of the lattice point problem, $P_{θ\mathcal{B}}(t)$, is of order $O_θ(t^{d-2+2/(d+1)-ζ_d})$ with a positive number $ζ_d$. Furthermore we extend the estimate of the above type, in the planar case, to general compact convex domains.

math.NT↗
arXiv2011

Lattice points in large convex planar domains of finite type

Let $\mathcal{B}$ be a compact convex planar domain with smooth boundary of finite type and $\mathcal{B}_θ$ its rotation by an angle $θ$. We prove that for almost every $θ\in[0, 2π]$ the remainder $P_{\mathcal{B}_θ}(t)$ is of order $O_θ(t^{2/3-ζ})$ with a positive number $ζ$ independent of the domain.

math.NT↗
arXiv2010

On lattice points in large convex bodies

We consider a compact convex body $\mathcal{B}$ in $\mathbb{R}^d$ $(d\geqslant 3)$ with smooth boundary and nonzero Gaussian curvature and prove a new estimate of $P_{\mathcal{B}}(t)$, the remainder in the lattice point problem, which improves previously known best result.

math.NT↗
arXiv2010

An ultra-cold, molecular Rydberg plasma with exceptionally long lifetime and strongly-coupled properties formed by threshold laser excitation in the expansion region of a supersonic jet

An ultra-cold molecular plasma with extraordinarily long lifetime (~0.5 ms) is generated under strong collisional cooling conditions in the expansion region of a seeded supersonic jet expansion close to the nozzle. A resonant two-photon one-color laser process excites para-difluorobenzene molecules into the high-n Rydberg threshold region. Disorder heating during plasma formation is quenched by the high collision rate in the expansion, which keeps the ions at the translational jet temperature of 0.2K-0.7K. The Coulomb coupling parameter \Gammai for the ions is expected to be ca. 230-820.

cond-mat.quant-gas↗
arXiv2010

Some properties of a long lifetime strongly-coupled molecular plasma produced by high Rydberg excitation of nitric oxide in a supersonic free jet

A long life-time (>0.3 ms) strongly-coupled molecular Rydberg plasma is generated by the excitation of nitric oxide into the high-n Rydberg threshold region in the high-density region of a supersonic jet expansion. After 310 \mus the plasma has expanded to a size of ca. 3 cm. When subjected to very small DC fields from 0.2 to 1.0 V/cm the plasma reveals a much smaller high-density core structure of only 0.6 cm. The molecular Rydberg plasma is observed over a broad range of excitation energies, from threshold down to Rydberg states as low as n = 19.

cond-mat.quant-gas↗