Questions tagged [laplacian]

The Laplacian matrix is the representation of a graph in matrix form.

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16
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221 views

Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?

Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...
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589 views

Are harmonic mappings non-singular outside a set of measure zero?

Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$. Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
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106 views

implicit function theorem and harmonic mapping

We are given two Riemannian manifolds $M,N$ of dimension $m$ and $n$ and a function $G \colon M \times N \to \mathbb{R}^n$ which satisfies the assumptions of the implicit function theorem, meaning ...
7
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353 views

Sequence of graphs with small $\lambda_1$ (the smallest nonzero eigenvalue of a regular finite graph)

The combinatorial laplacian on a finite graph $G$ can be defined as $ \Delta: \mathbb{C}^G \to \mathbb{C}^G$ sending the function $f:G \to \mathbb{C}$ to $(\Delta f)(v) = \sum_{v' \sim v} \big( f(v)-f(...
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271 views

Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett

( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .) Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
6
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221 views

Complex factorization of the angular part of the Laplacian

Some time ago some research led me to the following equality: \begin{equation} \frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...
6
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75 views

Density of squares of radial eigenfunctions

The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
6
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210 views

Locally symmetric spaces: spectrum of the Laplacian

Let $M = \Gamma\backslash X$ denote a locally symmetric space of non-compact type and $\Delta$ the Laplacian on $L^2(M)$. It is known that the spectrum of $\Delta$ decomposes into finitely many ...
6
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208 views

How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
5
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98 views

Laplace Beltrami eigenvalues on surface of polytopes

The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...
5
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139 views

Permanent bound for Laplacian matrix of signed graph

In 1986, Prof. RB Bapat shown that (see here) if $G$ is a simple connected graph on $n$ vertices, then, the permanent per$\big(L(G)\big)\ge 2(n-1)\kappa(G)$, where $L(G)$ is the Laplacian matrix of $G$...
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105 views

Different definitions of fractional sobolev spaces

Let $\Omega$ be a bounded and smooth domain in $\mathbb R^d$. For any $s\in (0,1)$ we can define $H_s(\Omega)$ to be the space of functions $u\in L^2(\Omega)$ such that $$(x,y)\mapsto \frac{|u(x)-u(y)|...
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160 views

Hilbert Modular Surface: Eigenfunctions of The Laplacian

The spectrum of the Laplacian on $L^2$ of a Hilbert modular surface decomposes into a discrete part and a continuous part $[1/4,\infty)$. The continuous part contains eigenvalues $\geq 1/4$. I would ...
5
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148 views

A conjecture on shape optimization for Dirichlet-Laplacian

For a given domain $G$, with sufficiently smooth boundary, in the plane we denote the first two eigenvalues of Dirichlet-Laplacian on $G$ of by $\lambda_1(G)$ and $\lambda_2(G)$. $\textbf{Open(?) ...
4
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72 views

Spectrum of Laplace-Beltrami with piecewise constant coefficients

By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...
4
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49 views

Is $\Delta \phi$ monotone operator on $H^1(\mathbb{R}^d)$ for monotone $\phi$

Let $H^1(\mathbb{R}^d)$ be the usual Sobolev space and let $\phi: \mathbb{R} \to \mathbb{R}$ be a non decreasing Lipschitz function with $\phi(0)=0$. Is the operator $\Delta \phi $ on $H^1(\mathbb{R}^...
4
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129 views

Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$

Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
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112 views

System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence

I'm looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands. Let $F$ be a totally real number field with $[F:\mathbb{Q}]=r$, let $B_1=M_2(F)$ be the split ...
4
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269 views

Eigenspaces of the Laplace operator on a unit ball

I am interested in structures of the eigenspaces of the Laplace operator on the $n$-dimensional unit ball with Neumann or Dirichlet boundary conditions as representations of the special orthogonal ...
4
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307 views

non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e., the matrix is not only weak diagonal-dominant, but ...
3
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71 views

Kronecker limit formula for antiperiodic boundary conditions

The celebrated Kronecker limit formula gives the $\zeta$-reguralized determinant of the Laplacian on the torus $\mathbb{R}/(\mathbb{Z}\omega_1+\mathbb{Z}\omega_2)$ in terms of Dedekind eta function of ...
3
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49 views

Tight bound on spectral gap of compact homogeneous manifold?

This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter $d$: $$ \lambda_1 \ge c/d^2, $$ where $c=\pi^2/4$. Can this bound be ...
3
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167 views

Laplace spectra of “half” grid graph

Let $G=(E,V)$ be a simple graph. The graph Laplacian is given by $$ L= D-A,$$ where $D$ is the degree matrix (diagonal matrix with entries corresponding to the degree of the vertex) and $A$ the ...
3
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120 views

Differential operators on a compact Lie group associated to bracket-generating sets

Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$. Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$. Assume that $\{X_1,\dots,X_h\}$ is ...
3
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95 views

The heat kernel in Hermitian bundles over Riemannian manifolds

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....
3
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74 views

Estimate a function given an estimate of its Laplacian

Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions: \begin{eqnarray*} \int_B |f_\lambda(x)|^2dx\leq 1,...
3
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92 views

Error term in the Euclidean Weyl law

Let $\Omega\subset \mathbb R^n$ be an open bounded set with smooth boundary. The Laplacian on $\Omega$ with Dirichlet boundary conditions has discrete spectrum $\lambda_1\le \lambda_2\le \ldots$ that ...
3
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50 views

Gradient of spectral function on noncompact homogeneous space

Let $(M,g)$ be a noncompact Riemannian manifold whose isometry group acts transitively on $M$, i.e. a (not necessarily normal) homogeneous space. Let $e_{\lambda}(x,y)$ be the integral kernel of $f \...
3
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353 views

Spectrum of the Laplace-Beltrami operator on a domain of finite volume in the hyperbolic space $H^n$

What is known about the ($L^2$) spectrum of the minus Laplace-Beltrami operator ($- \Delta$) with zero boundary conditions on $B =H^n/\Gamma$, where $H^n$ is $n$-dimensional hyperbolic space ($n>1$...
3
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213 views

Laplace-Beltrami of the Gauss map

Let $M$ be a surface in $\mathbb{R}^3$ given by a regular chart, say $X:M \longrightarrow \mathbb{R}^3$, with its first fundamental form $g$, Gauss map $N$, Gaussian curvature $K$ and mean curvature $...
3
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139 views

Spectral theory of differential forms over a circle bundle

Here is the set up : I consider the unitary tangent bundle of a surface $(S,g)$ endowed with the Sasakian metric ; $(T^1S, g_s)$, in fact we have the following fibration : \begin{equation*} \mathbb{...
3
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0answers
152 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
3
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0answers
122 views

What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
3
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0answers
119 views

Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$). Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$. ...
2
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0answers
71 views

Spectrum of the Witten Laplacian on compact Riemannian manifolds

Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$ How generally is it true that this ${\rm ...
2
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179 views

Laplacian on a manifold with two boundary components

I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...
2
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90 views

May the heat kernel of a connection Laplacian vanish?

Let $M$ be a Riemannian manifold and $E \to M$ be a Hermitian bundle. If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs ...
2
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60 views

Is it possible to compute a valid Laplacian matrix from an effective resistance matrix?

I am wondering whether it is possible to retrieve a node-admittance matrix $G$ (also called Laplacian matrix) in a purely resistive network composed of nets $\{1, \dots, i, \dots, j, \dots, n\}$, from ...
2
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0answers
84 views

Li-Yau inequality $\frac{4\pi}{|D|} k<\lambda_{k}<\frac{4\pi}{|D|} k+c\sqrt{k}$ for large enough k

For proving another interesting question: Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $ I need the following inequality for Dirichlet ...
2
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0answers
350 views

Fractional sobolev spaces

On the whole space $\mathbb R^d$, the fractional Sobolev space $H_s(\mathbb R^d)$ of order $s\in \mathbb R$ can be defined as the subspace of tempered distributions $T$ such that $\mathcal F T \in L^...
2
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145 views

Inequalities concerning principal eigenvalues of Laplacian with different boundary conditions

Suppose $\Omega\subset\mathbb R^2$ is a bounded simply connected domain with sufficiently smooth boundary. Consider the following three BVPs (respectively with Dirchlet, Neumann and certain non-local ...
2
votes
0answers
56 views

First eigenvalue for strictly convex domains

Let $M^n$ be a compact Riemannian manifold with boundary, suppose 1). $Ric(M)\ge (n-1)$ and 2). the principle curvatures of the boundary is bounded from below by $h\ge 0$. Is there any results on the ...
2
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0answers
214 views

Ellipticity of Bott-Chern Laplacian

I want to prove that Bott-Chern Laplacian $$\tilde{\Delta}_{BC}^{p,q}=(\partial\bar\partial)(\partial\bar\partial)^*+(\partial\bar\partial)^*(\partial\bar\partial)+(\bar\partial^*\partial)(\bar\...
2
votes
0answers
185 views

Estimate for the first eigenvalue of the Laplacian

I was studying the paper of S. T. Yau - Seminar on Differential Geometry - and there asks if the first eigenvalue is equal to $ n $, if we have a embedded oriented Riemannian manifold and closed ...
2
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0answers
511 views

Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix

I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...
2
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0answers
84 views

Obstructions for existence of a Riemannian metric such that a given function is harmonic

Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic ...
2
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0answers
409 views

Is Laplacian a surjective operator?

For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image ...
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0answers
44 views

What is a random eigenfunction on the hyperbolic plane?

Is there an (invariant under isometries) notion of a random eigenfunction on the hyperbolic plane, for a given eigenvalue? It is a reference request because the answer is probably positive and I even ...
1
vote
0answers
102 views

Laplacian of Fourier-like function

The question that follows has to do with the effects of a turbulent atmosphere on wave propagation. The structure function, $D(\vec{r})$, which is defined as, \begin{equation} D(\vec{r}) = \left\...
1
vote
1answer
82 views

Relation between expected values of eigenvalues of Laplacian matrix of a graph and eigenvalues of expected Laplacian matrix of that graph?

Particularly, I am dealing with Erdős–Rényi random 𝐺(𝑛,𝑝), so the expected Laplacian matrix of 𝐺(𝑛,𝑝) is 𝑝(𝐽𝑛−𝐼𝑛), where 𝐽𝑛 and 𝐼𝑛 are one and identity matrices, respectively. In ...