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Zuoqin Wang

Publications and source records attributed to Zuoqin Wang.

At least 19 recordsLinked to original sources
arXiv2026

Equivariant Morse theory for Lie algebra actions on Riemannian foliations

We consider a transverse isometric action of a finite-dimensional Lie algebra $\mathfrak g$ on a Riemannian foliation. In this setting, we study equivariant Morse--Bott theory on the leaf space of the foliation. Among other results, we establish a foliated Morse--Bott lemma for $\mathfrak g$-invariant basic Morse--Bott functions and a foliated analogue of the usual handle presentation theorem. In the non-equivariant case, we use these results to give a new proof of the Morse inequalities for Riemannian foliations. In the equivariant case, we apply them to Hamiltonian actions of abelian Lie algebras on presymplectic manifolds whose underlying foliations are Riemannian, and we extend the Kirwan surjectivity and injectivity theorems from equivariant symplectic geometry to this setting. As a consequence, Kirwan surjectivity and injectivity hold for Hamiltonian torus actions on symplectic orbifolds.

math.DG↗
arXiv2026

The Steklov Determinant and Compactness of Isospectral Planar Domains

We prove that every Steklov isospectral family of compact smooth planar domains is compact in the $C^\infty$ topology, answering an open question of Colbois, Girouard, Gordon, and Sher. The proof has two main parts. First, a trace comparison principle for Dirichlet-to-Neumann operators yields monotonicity of negative Steklov zeta values and compactness within a fixed conformal class for genus-zero flat surfaces. Second, we analyze the normalized Steklov determinant on degenerating hyperbolic surfaces with geodesic boundary. Its asymptotics are expressed in terms of shrinking boundary components and small Neumann and Dirichlet eigenvalues, which in genus zero are compared with weighted graph Laplacians. This rules out degeneration under a bound on the total hyperbolic boundary length. We finally establish this bound for planar domains by geometric non-collapse estimates.

math.SP↗
arXiv2026

Steklov Spectral Uniqueness of Geodesic Balls in 3-Dimensional Space Forms via Guillemin--Wodzicki Residues

Let $Λ$ be the Dirichlet-to-Neumann operator on the boundary of a compact three-dimensional Riemannian manifold. Using the Lee--Uhlmann full-symbol formula and Weyl's invariant theory, we compute the Guillemin--Wodzicki residues of $Λ$ and $Λ^2$ explicitly, and hence the first two logarithmic coefficients in the Steklov heat trace. As an application, we prove that geodesic balls in simply connected three-dimensional space forms are determined by their Steklov spectra among smooth domains in the same space form. More generally, we obtain a rigidity theorem in the class of compact constant-curvature manifolds with smooth, not necessarily connected, boundary. In the Euclidean case, we also prove spectral uniqueness for concentric spherical shell regions.

math.SP↗
arXiv2026

Spectral and Geometric Stability for the Reciprocal Sum of Neumann Eigenvalues

We establish quantitative stability for the reciprocal-sum isoperimetric inequality for the first $d$ nonzero Neumann eigenvalues, recently proved by He, Li, and Tang. We prove that for bounded Lipschitz domains in $\mathbb{R}^d$, the reciprocal-sum deficit controls quadratically both the normalized eigenvalue displacements and the Fraenkel asymmetry, while controlling the displacement of the eigenvalue sum linearly. A further result shows that the classical Szegő--Weinberger deficit controls both the gap between the first two nonzero eigenvalues and the full width of the first eigenvalue cluster. Nearly spherical perturbations demonstrate that all the stability exponents are optimal. In dimension two, the same matrix method yields improved constraints on the joint spectral image of the first two nonzero eigenvalues.

math.SP↗
arXiv2026

Steklov Spectral Geometry for Annular Surfaces: Inverse spectral results and isospectral compactness

We study the inverse spectral problems for the Steklov spectrum on compact surfaces with boundary. We prove that among flat annular surfaces, the lateral surface of a conical frustum is uniquely determined by its Steklov spectrum. As a consequence, each circular annulus is uniquely determined among all planar domains, providing the first example of a non-simply connected Euclidean domain with this property. Furthermore, we show that any family of Steklov isospectral flat annular surfaces is compact in the $C^\infty$ topology, extending previous results for simply connected planar domains. These results are established through a detailed analysis of the spectral zeta function and the zeta-regularized determinant associated with the Dirichlet-to-Neumann operator for annular surfaces.

math.SP↗
arXiv2025

Pólya's conjecture for thin products

Let $Ω\subset \mathbb R^d$ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue $λ_k(Ω)$ and its Neumann eigenvalue $μ_k(Ω)$ have the same leading asymptotics $w_k(Ω)=C(d,Ω)k^{2/d}$ as $k \to \infty$. G. Pólya conjectured in 1954 that each Dirichlet eigenvalue $λ_k(Ω)$ is greater than $w_k(Ω)$, while each Neumann eigenvalue $μ_k(Ω)$ is no more than $w_k(Ω)$. In this paper we prove Pólya's conjecture for thin products, i.e. domains of the form $(aΩ_1) \times Ω_2$, where $Ω_1, Ω_2$ are Euclidean domains, and $a$ is small enough. We also prove that the same inequalities hold if $Ω_2$ is replaced by a Riemannian manifold, and thus get Pólya's conjecture for a class of ``thin" Riemannian manifolds with boundary.

math.SP↗
arXiv2025

Prescription of the Robin spectrum

Let $M$ be a compact connected smooth manifold with smooth boundary, and let $ρ$ be a positive continuous function on the boundary which is served as the Robin parameter. In this paper, we study three problems concerning the prescription of finite Robin spectrum: (1) Prescribing finitely many Robin eigenvalues and the volume. (2) Within a fixed conformal class, prescribing the multiplicities of finitely many Robin eigenvalues. (3) Within a fixed conformal class, prescribing finitely many distinct Robin eigenvalues and the volume. A key step in our method is to solve the corresponding problems for the Dirichlet spectrum. As a consequence, we also obtain analogous results for the Dirichlet case.

math.SP↗
arXiv2024

Bessel functions and Weyl's law for balls and spherical shells

The purpose of this paper is twofold. One is to investigate the properties of the zeros of cross-products of Bessel functions or derivatives of ultraspherical Bessel functions, as well as the properties of the zeros of the derivative of the first-kind ultraspherical Bessel function. The properties we study include asymptotics (with uniform and nonuniform remainder estimates), upper and lower bounds and so on. In addition, we provide the number of zeros of a certain cross-product within a large circle and show that all its zeros are real and simple. These results may be of independent interest. The other is to investigate the Dirichlet/Neumann Laplacian on balls and spherical shells in $\mathbb{R}^d$ ($d\geq 2$) and the remainder of the associated Weyl's law. We obtain new upper bounds in all dimensions, both in the Dirichlet and Neumann cases. The proof relies on our studies of Bessel functions and the latest development in the Gauss circle problem, which was driven by the application of the emerging decoupling theory of harmonic analysis.

math.CA↗
arXiv2023

Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum

In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold $M$ of dimension $n\geq 3$ with a non-empty smooth boundary $\partial M$. We show that for any finite increasing sequence of real numbers $0<a_1<a_2 \leq a_3 \leq \cdots \leq a_N$ and any positive number $V$, there exists a Riemannian metric $g$ on $M$ such that $\mathrm{Vol}(M,g)=V$ and $λ^\mathcal{D}_k(M,g)=a_k$ for any integer $1 \leq k \leq N$.

math.DG↗
arXiv2021

Integral representations of isotropic semi-classical functions and applications

In \cite{GUW} we introduced a class of "semi-classical functions of isotropic type", starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the "oscillatory functions of Lagrangian type" that have played major role in semi-classical and micro-local analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual clean-intersection hypothesis). The simplest examples of isotropic states are the "coherent states", a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrödinger in the 1920's. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact. We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary \ref{cor:altWeyl}).

math.AP↗
arXiv2020

Inverse spectral results for non-abelian group actions

In this paper we will extend to non-abelian groups inverse spectral results, proved by us in an earlier paper, for compact abelian groups, i.e. tori. More precisely, Let $\mathsf G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $X$. We will show that for the Schrödinger operator $-\hbar^2 Δ+V$ with $V \in C^\infty(X)^{\mathsf G}$, the potential function $V$ is, in some interesting examples, determined by the $\mathsf G$-equivariant spectrum. The key ingredient in this proof is a generalized Legendrian relation between the Lagrangian manifolds $\mathrm{Graph}(dV)$ and $\mathrm{Graph}(dF)$, where $F$ is a spectral invariant defined on an open subset of the positive Weyl chamber.

math.SP↗
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

Spectral properties of semi-classical Toeplitz operators

The main results of this paper are an asymptotic expansion in powers of $\hbar$ for the spectral measure $μ_\hbar$ of a semi-classical Toeplitz operator, $Q_\hbar$, and an equivariant version of this result when $Q_\hbar$ admits an $n$-torus as a symmetry group. In addition we discuss some inverse spectral consequences of these results.

math.SP↗
arXiv2017

On Isospectral compactness in conformal class for 4-manifolds

Let $(M, g_0)$ be a closed 4-manifold with positive Yamabe invariant and with $L^2$-small Weyl curvature tensor. Let $g_1 \in [g_0]$ be any metric in the conformal class of $g_0$ whose scalar curvature is $L^2$-close to a constant. We prove that the set of Riemannian metrics in the conformal class $[g_0]$ that are isospectral to $g_1$ is compact in the $C^\infty$ topology.

math.SP↗
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↗
arXiv2016

Semiclassical states associated to isotropic submanifolds of phase space

We define classes of quantum states associated to isotropic submanifolds of cotangent bundles. The classes are stable under the action of semiclassical pseudo-differential operators and covariant under the action of semiclassical Fourier integral operators. We develop a semiclassical symbol calculus for them; the symbols are symplectic spinors. We outline various applications.

math.AP↗