Laplace-Beltrami Operator Research Articles

Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tori

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on “curved” NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get Lp-versions of the usual formula for the trace of ΨDOs on NC tori, i.e., for combinations of ΨDOs with Lp-position operators. Next, we get Lp-versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a. Dixmier traces) of products of Lp-position operators with ΨDOs or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators associated with powers of Laplace-Beltrami operators and Lp-potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for our classes of operators on curved NC tori. This supersedes a previous conjecture of [63].

Jacobi partial waves for a set of 3D noncentral rational scatterers

The common tool of choice for basis expansions for the scattering problem with 3D quantum systems remains the spherical harmonics as eigenfunctions of the Laplace–Beltrami operator on the sphere, with approximations for deviations made around the usually dominant s-wave spherically symmetric state. However, with the growing number of technologically accessible nonspherically symmetric geometries of cold atomic and molecular systems, there is a need to explore as orthonormal bases for partial wave analysis the larger class of weighted Jacobi polynomials, subsuming the spherical harmonics. In particular, the lowest angular state for this bigger class is a toroid instead of a spherical s-orbital. This allows analytic treatment of a wider array of rational angular-dependent potentials which can describe rings and systems with topological constraints such as monopoles. Here, we present exact analytic solutions for the quantum scattering problem by partial wave analysis using the weighted Jacobi polynomials as an expanded basis. We obtain the scattering amplitude, differential and total cross-sections for exactly solvable 3D potentials included in the Smorodinsky-Winternitz noncentral systems with dynamical symmetries. Moreover, this procedure also solves the quantum scattering problem from a novel series of rational trigonometric forms of anisotropic potentials including double ring-shaped configurations.

Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions

We prove a quadratic sparse domination result for general non-integral square functions S. That is, for p_0 in [1,2) and q_0 in (2,infty ], we prove an estimate of the form where q_{0}^{*} is the Hölder conjugate of q_{0}/2, M is the underlying doubling space and {mathcal {S}} is a sparse collection of cubes on M. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace–Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space L^{p}(w).

NOTES ON CONFORMAL PERTURBATION OF HEAT KERNELS

Let [Formula: see text] be a smooth n-dimensional Riemannian manifold for [Formula: see text]. Consider the conformal perturbation [Formula: see text] where [Formula: see text] is a smooth bounded positive function on [Formula: see text]. Denote by [Formula: see text] the Laplace–Beltrami operator of manifold [Formula: see text]. In this paper, we derive the upper bounds of the heat kernels for [Formula: see text] with [Formula: see text]. Moreover, we also investigate the gradient estimates of the heat kernel for [Formula: see text].

Essential Self-Adjointness of Klein-Gordon Type Operators on Asymptotically Static, Cauchy-Compact Spacetimes

Let $X=\mathbb{R}\times M$ be the spacetime, where $M$ is a closed manifold equipped with a Riemannian metric $g$, and we consider a symmetric Klein-Gordon type operator $P$ on $X$, which is asymptotically converges to $\partial_t^2-\triangle_g$ as $|t|\to\infty$, where $\triangle_g$ is the Laplace-Beltrami operator on $M$. We prove the essential self-adjointness of $P$ on $C_0^\infty(X)$. The idea of the proof is closely related to a recent paper by the authors on the essential self-adjointness for Klein-Gordon operators on asymptotically flat spaces.

Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

This work studies the spectral convergence of graph Laplacian to the Laplace-Beltrami operator when the graph affinity matrix is constructed from $N$ random samples on a $d$-dimensional manifold embedded in a possibly high dimensional space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove that, with Gaussian kernel, one can set the kernel bandwidth parameter $\epsilon \sim (\log N/ N)^{1/(d/2+2)}$ such that the eigenvalue convergence rate is $N^{-1/(d/2+2)}$ and the eigenvector convergence in 2-norm has rate $N^{-1/(d+4)}$; When $\epsilon \sim (\log N/N)^{1/(d/2+3)}$, both eigenvalue and eigenvector rates are $N^{-1/(d/2+3)}$. These rates are up to a $\log N$ factor and proved for finitely many low-lying eigenvalues. The result holds for un-normalized and random-walk graph Laplacians when data are uniformly sampled on the manifold, as well as the density-corrected graph Laplacian (where the affinity matrix is normalized by the degree matrix from both sides) with non-uniformly sampled data. As an intermediate result, we prove new point-wise and Dirichlet form convergence rates for the density-corrected graph Laplacian. Numerical results are provided to verify the theory.

Laplacian and Brownian motion on positive definite matrices, revisited

We present explicit expression for the Laplace–Beltrami operator on the symmetric space of positive definite matrices by using the Iwasawa decomposition and study the corresponding diffusion process, the Brownian motion. We show that some inductive arguments in the dimensions work.

Alignment of highly resolved time-dependent experimental and simulated crash test data

We investigate for car and component crash tests the comparison of highly resolved experimental data with corresponding simulation data. Due to recent advances for optical measurement systems, one can nowadays obtain surface measurement data from a real crash experiment with high resolution in space and time. These advances call for new data processing methods that allow an alignment of this experimental data with numerical simulation results. We propose an approach based on a data representation stemming from a discrete Laplace–Beltrami operator, which allows such an alignment as well as a joint visual comparative analysis of both data sources. The method enables the identification of the best corresponding simulation among several numerical results, which also allows inferring physical quantities that cannot be measured in experiments. We evaluate the procedure on synthetic and real experimental data from two different crashworthiness setups.

Hypersphere Having ΔIIx = Ax in E4

We consider hypersphere x = x(u, v, w) in the four dimensional Euclidean space. We calculate the Gauss map, and the curvatures of it. Moreover, we compute the second Laplace-Beltrami operator the hypersphere satisfying ΔIIx = Ax, where A ϵ Mat (4,4).

Numerical approximation of the square phase-field crystal dynamics on the three-dimensional objects

In the natural world, the phase field transition on the surface of a three-dimensional (3D) object is common. The square phase-field crystal (SPFC) equation is an effective model for describing the phase transition from liquid to solid with lattice structure. This study aims to present an efficient and practical computational scheme to simulate the SPFC dynamics on arbitrary surfaces. The surface is represented by the zero level-set of a singed distance function. The closest point method and appropriate pseudo-Neumann boundary condition are used to make the numerical solution to be constant along the normal direction to the arbitrary surface. In this sense, the calculations are transformed into a 3D discrete narrow computational band domain embedding the arbitrary surface and the finite difference method (FDM) is adopted to discretize the governing equation in 3D space. To maintain second-order spatial accuracy, the standard seven-point discrete Laplacian operator is applied to replace the discrete surface Laplace–Beltrami operator. We treat the nonlinear term in a totally explicit form and utilize the stabilization technique to suppress the stiffness of the numerical scheme. Therefore, we avoid solving any nonlinear equation or linear equation with variable coefficients. In each time step, we only need to update the linear system with constant coefficients. The proposed numerical scheme is highly efficient and its implementation is simple. Various numerical experiments demonstrate that the proposed numerical scheme not only achieves the expected accuracy but also works well for the simulations of phase transition and crystallization on surfaces with smooth and sharp shapes.

Norms of certain functions of a distinguished Laplacian on the $$ax+b$$ groups

The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator \({\mathcal {L}}\) on the ‘\(ax+b\)’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type \(\psi (\sqrt{{\mathcal {L}}})\exp (it \sqrt{{\mathcal {L}}})\), with \(\psi \in C_0({\mathbb {R}})\). We show that for \(t\rightarrow +\infty \), the convolution kernel \(k_t\) of this operator satisfies $$\begin{aligned} \Vert k_t\Vert _1\asymp t, \qquad \Vert k_t\Vert _\infty \asymp 1, \end{aligned}$$so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of \({\mathcal {L}}\) and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator \({\tilde{\Delta }}\), closely related to \({\mathcal {L}}\). The functions include in particular \(\exp (-t{\tilde{\Delta }}^\gamma )\), \(t>0,\gamma >0\), and \(({\tilde{\Delta }}-z)^s\), with complex z, s.

Product formulas and convolutions for Laplace–Beltrami operators on product spaces: beyond the trivial case

Product formulas and convolutions for Laplace–Beltrami operators on product spaces: beyond the trivial case

Generalized Helical Hypersurfaces Having Time-like Axis in Minkowski Spacetime

In this paper, the generalized helical hypersurfaces x=x(u,v,w) with a time-like axis in Minkowski spacetime E14 are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of x are calculated. Moreover, the curvatures of the generalized helical hypersurface x are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of x are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in E14.

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.

Data-Driven Fuzzy Transform

The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace–Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size.

Improved spectral convergence rates for graph Laplacians on ε-graphs and k-NN graphs

In this paper we improve the spectral convergence rates for graph-based approximations of weighted Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity ε, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of a weighted Laplace-Beltrami operator at a rate of O(n−1/(m+4)), up to log factors, where m is the manifold dimension and n is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include ε-graphs and k-NN graphs. We also present the results of numerical experiments analyzing convergence rates on the two dimensional sphere.

Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure

We study the covariant derivatives of an eigenfunction for the Laplace–Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.

Fractional Klein–Gordon equation on AdS2+1

We propose a covariant definition of the fractional Klein–Gordon equation (KGE) with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional KGE on AdS2+1, computing the explicit kernel representation of the fractional Laplace–Beltrami operator as well as the two-point propagator of the fractional KGE. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information.

On the pointwise converse of Fatou’s theorem for Euclidean and real hyperbolic spaces

In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space ℝ n+1+ . We show that similar results remain valid for more general approximate identities. We apply this result to prove a result regarding boundary behavior of certain nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn. We shall also prove a generalization of a result regarding large time behavior of a solution of the heat equation proved in [17]. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn.

Data-driven modeling with fuzzy sets and manifolds

The manifold hypothesis states that the shape of the observed data is relatively simple and that it lies on a low-dimensional manifold embedded in a high-dimensional space. We contribute to the problem of data-driven modeling by treating it as an inverse problem where the model defines a Euclidean space with a Riemannian manifold structure. In particular, our contribution shows that a fuzzy set on a bounded support defines a Riemannian manifold that can be embedded in a multidimensional space where dimension is a model parameter. A noticeable advantage of the proposed approach is its connection with the values of the membership function and independence from the dimension of the data being modeled.Last but not least, we have found various formal representations of the Laplace-Beltrami operator and use its values as an estimate of the quality of the approximate solution to the inverse problem.