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Siqi Fu

Publications and source records attributed to Siqi Fu.

At least 19 recordsLinked to original sources
arXiv2026

Multiplicities of eigenvalues and quadratic representations of integers

We study the set $M$ of all multiplicities of non-zero eigenvalues for the Laplace operator on a two-dimensional rectangle or torus. We show that for a rectangle with the side length ratio $r$, $M=\mathbb{N}$, the set of all positive integers, if and only if $r^2$ is rational. For a torus whose generating vectors have a length ratio $r$ and the angle between them $\theta$, we show that $M$ is an infinite set if and only if both $r\cos\theta$ and $r^2$ are rational. In this case, $M=2\mathbb{N}$, $4\mathbb{N}$, or $6\mathbb{N}$, and we obtain a characterization for each of these cases in term of $r\cos\theta$ and $r^2$. In the case when at least one of $r\cos\theta$ or $r^2$ is irrational, we show that $M=\{2\}$ or $\{2, 4\}$, and obtain a characterization for these cases. We prove these results by studying the number of integral lattice points on dilated ellipses.

math.NT↗
arXiv2019

Spectral Stability of the $\bar\partial-$Neumann Laplacian: Domain Perturbations

We study spectral stability of the $\bar\partial$-Neumann Laplacian on a bounded domain in $\mathbb{C}^n$ when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $\bar\partial$-Neumann Laplacian on bounded pseudoconvex domains in $\mathbb{C}^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property ($P$), and quantitative estimates on smooth bounded pseudoconvex domains of finite D'Angelo type in $\mathbb{C}^n$.

math.CV↗
arXiv2017

Hearing pseudoconvexity in Lipschitz domains with holes via $\overline\partial$

Let $Ω=\widetildeΩ\setminus \overline{D}$ where $\widetildeΩ$ is a bounded domain with connected complement in $\mathbb C^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of $\widetildeΩ$ with connected complement in $\widetildeΩ$. We obtain characterizations of pseudoconvexity of $\widetildeΩ$ and $D$ through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of $\widetildeΩ$ and $D$ are Lipschitz and $C^2$-smooth respectively, then both $\widetildeΩ$ and $D$ are pseudoconvex if and only if $0$ is not in the spectrum of the $\overline\partial$-Neumann Laplacian on $(0, q)$-forms for $1\le q\le n-2$ when $n\geq 3$; or $0$ is not a limit point of the spectrum of the $\overline\partial$-Neumannn Laplacian on $(0, 1)$-forms when $n=2$.

math.CV↗
arXiv2012

Stability of the Bergman kernel on a tower of coverings

We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective version of Rhodes' result is given for a tower of coverings on a compact Riemann surface of genus greater than or equal to 2. Stability of the Bergman kernel is established for towers of coverings on hyperbolic Riemann surfaces and on complete Kaehler manifolds satisfying certain potential conditions. As a consequence, stability of the Bergman kernel is established for any tower of coverings of Riemann surfaces when the top manifold is simply-connected.

math.CV↗
arXiv2010

Positivity of the $\bar\partial$-Neumann Laplacian

We study the $\bar\partial$-Neumann Laplacian from spectral theoretic perspectives. In particular, we show how pseudoconvexity of a bounded domain is characterized by positivity of the $\bar\partial$-Neumann Laplacian.

math.CV↗
arXiv2010

Comparison of the Bergman and Szegö kernels

The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in ${\mathbb C}^n$ is bounded from above by $δ|\logδ|^p$ for any $p>n$, where $δ$ is the distance to the boundary. For a class of domains that includes those of D'Angelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by $δ|\logδ|^p$ for any $p<-1$. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of $δ$.

math.CV↗
arXiv2007

The $\overline\partial$-cohomology groups, holomorphic Morse inequalities, and finite type conditions

We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if and only if for any positive constant $C$, the number of eigenvalues of the $\overline\partial$-Neumann Laplacian less than or equal to $Ck$ grows polynomially as $k$ tends to infinity.

math.CV↗
arXiv2005

Spectrum of the d-bar-Neumann Laplacian on polydiscs

The spectrum of the d-bar-Neumann Laplacian on a polydisc in several complex variables is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of infinite multiplicity.

math.CV↗
arXiv2004

Hearing pseudoconvexity with the Kohn Laplacian

A bounded domain in several complex variables with connected Lipschitz boundary is pseudoconvex if and only if the bottom of the (essential) spectrum of the Kohn Laplacian is positive on all (0, q)-forms with square-integrable coefficients.

math.CV↗
arXiv2003

Compactness in the d-bar Neumann problem, magnetic Schrodinger operators, and the Aharonov-Bohm effect

Compactness of the d-bar Neumann operator is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth example is constructed in which condition (P) fails to hold, yet the Neumann operator is compact. The main result, in contrast, is that for smoothly bounded Hartogs domains, condition (P) of Catlin and Sibony is equivalent to compactness. The analyses of both compactness and condition (P) boil down to properties of the lowest eigenvalues of certain sequences of Schrodinger operators, with and without magnetic fields, parametrized by a Fourier variable resulting from the Hartogs symmetry. The nonsmooth counterexample is based on the Aharonov-Bohm phenomenon of quantum mechanics. For smooth domains, we prove that there always exists an exceptional sequence of Fourier variables for which the Aharonov-Bohm effect is quite weak. This sequence can be quite sparse, so that the failure of compactness is due to a rather subtle effect.

math.CV↗
arXiv1997

Compactness of the d-bar-Neumann problem on convex domains

The d-bar-Neumann operator on (0,q)-forms ($1\le q \le n$) on a bounded convex domain Omega in C^n is compact if and only if the boundary of Omega contains no complex analytic (equivalently: affine) variety of dimension greater than or equal to q.

math.CV↗