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Questions tagged [laplacian]

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Lower bound on the nonzero Laplacian eigenvalue with the smallest real part

Consider a directed graph with $n$ vertices. The graph is not assumed to be connected, and therefore the multiplicity of the eigenvalue 0 may be greater than 1. I am looking for a nonzero lower bound ...
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0answers
65 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 ...
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171 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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107 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 ...
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66 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....
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1answer
244 views

How does the high-dimensional combinatorial Laplacian work?

When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C_k(X)$ is $$\partial_k([v_0,...,v_k])=\sum_{i=0}^k (-1)^i[v_0,...,...
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0answers
38 views

First eigenvalue for domains in hyperbolic space

I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...
9
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1answer
337 views

Bochner formula in different forms

I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only). For example I will need the following formula:...
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44 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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1answer
100 views

Conditions for a matrix to be a Graph Laplacian [closed]

Let M be a symmetric non-negative definite $n\times n$ matrix. Let $K_n$ denote the complete graph on $n$ vertices. Under what conditions is it possible to assign edge weights to $K_n$ in such a way ...
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3answers
180 views

Reference on spectral fractional Laplacian

Are there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome.
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61 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 $...
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2answers
164 views

How do eigenvalues of combinatorial Laplacian relates to automorphisms in graphs?

Is there a relation between eigenvalues of the graph Laplacian and the automorphism group of a simple graph? How are the multiplicities of Laplacian eigenvalues related to the order of the ...
2
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0answers
68 views

Asymptotic behavior of the Dirichlet-Laplacian eigenvalues [closed]

I found in a math book http://www.cambridge.org/dz/academic/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-partial-differential-equations?format=PB&...
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0answers
73 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,...
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1answer
100 views

Realization of symbol of Laplace operator via certain integral

Is there an elliptic operator $D$ on $C^{\infty}(S^2)$ whose principal symbol is not identical to thats of Laplacian but it satisfies $\int_{S^2} fDf =\int_{S^2} f\Delta (f)$ for all $f\in C^{\infty}(...
3
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1answer
90 views

Gaps in the spectrum of Laplace-Beltrami operators

Let us consider $\mathbb S^d$ the unit Euclidean sphere of $\mathbb R^{d+1}$ and let $\Delta_{\mathbb S^d}$ be the Laplace operator on $\mathbb S^d$. We have $$ -\Delta_{\mathbb S^d}=\sum_{k\in \...
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0answers
91 views

Laplacian on squashed spheres

Is anything known about the Laplacian on squashed spheres $S^{2n-1}_\omega$, where the ambient $C^n$ coordinates satisfy $$ 1= \sum_{i=1}^n \omega_i |z_i|^2 $$ for fixed real numbers $\omega_i$? for ...
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1answer
122 views

Resolvent of the Laplacian as a pseudodifferential operator and its single layer potential

In M.Taylor's book "Partial differential equations II" it is shown that the fundamental solution $E(x,y)$ of the Laplacian equation gives rise to an elliptic pseudodifferential operator $S$ on the ...
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0answers
94 views

Curvature of the boundary vs. normal derivative of the first eigenfunction

Disclaimer. I posted this question in Math.SE, but it haven't received enough attention. Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$....
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1answer
298 views

Can a harmonic vector field possess a limit cycle?

Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)? Note that the Laplacian of a vector field is defined via natural correspondence ...
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85 views

Eigenvalues of the Dirichlet Laplacian on an infinite strip

I would like to find the spectrum of the Laplacian on the infinite strip $(x,y)\in [0,L]\times\mathbb{R}$. So, I am looking for all solution of $$\frac{\partial^2}{\partial x^2} f(x,y) + \frac{\...
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0answers
125 views

Is Varadhan's formula valid for all pairs of points?

Most formulations of Varadhan's formula $$\lim _{t \to 0_+} 4t \log p_t(x,y) = -d(x,y)^2$$ that I have encountered do not specify where $(x,y)$ lives, so until today I imagined that $(x,y) \in M \...
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104 views

hyperbolic “Green function” on a product of upper half-planes

Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent $$ R(s)=(...
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0answers
177 views

domain of the Dirichlet Laplacian

Let $s\geq 0$ be a real number, $\Omega$ bounded and smooth domain in $\mathbb R^d$. We can define the spectral power of the Dirichlet Laplacian $(-\Delta_\Omega)$ on $\Omega$. Then, on the whole ...
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2answers
246 views

Interpolation spaces

In this paper, the authors claim that for $s\in [0,1]$, $\left [ H_0^1(\Omega), L^2(\Omega)\right ]_\theta=\text{dom}(-\Delta)^{\frac s 2}$, where $\Omega$ is a smooth and bounded domain in $\mathbb R^...
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0answers
69 views

introduction textbook to the Laplacian on a circle bundle

I am looking for an introduction to spectral theory of $\Delta$ on a circle bundle over a compact M. Is there an analog of Selberg trace formula?
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1answer
77 views

Positive analytic semigroup

On $L^p(\mathbb{R}^+)$, we consider the following operator: $$ Af:= f'',\qquad D(A):=\{u\in W^{2,p}(\mathbb{R}^+),u'(0)=0\} $$ Now i want to know if this operator is a generator of a positive analytic ...
4
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1answer
303 views

Laplace-Beltrami and the isometry group

H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$, $$ \Delta=\partial_i^2 $$ It is easy to prove that the most general differential operator that commutes with rotations and translations is ...
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0answers
72 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 ...
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127 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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2answers
323 views

General questions on the eigenfunctions of Laplacian and Dirac operators

We know that the eigenvalues of the Laplacian contains a lot of information of a Riemannian manifold, but they do not determine the full information ( Hearing the shape of a drum). And the ...
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0answers
77 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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0answers
293 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^...
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0answers
69 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 ...
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1answer
414 views

is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?

For two regular graphs $G$ and $H$, it is possible for them to share the same adjacent spectrum and the same laplacian spectrum. While, on the other hand, is it possible to have two non-regular graphs ...
3
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1answer
200 views

If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$?

Let me be more specific: If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$? The graph corresponding to $A$ is a directed ring, which is strongly connected and $1_n$ and $...
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0answers
56 views

For what potentials is the heat operator with a potential term hypoelliptic?

If $(M,g)$ is a Riemannian manifold and $\Delta$ is the Laplace-Beltrami (negatively defined) operator, is it possible to describe the class of smooth potentials $V :M \to \Bbb R$ that make the heat ...
3
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1answer
243 views

How is the Jacobian or Sandpile group of a graph computed?

From what I understand, given a graph, the Jacobian group and the Sandpile group refer to the same object. Until now, I have been computing this group in the way detailed in Chapter 1 of this ...
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0answers
110 views

Boundary regularity of the solution of a Poisson equation in a polyhedron

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be bounded and open $f\in L^2(\Lambda,\mathbb R^d)$ $u\in H_0^1(\Lambda,\mathbb R^d)$ with $$-\langle\nabla\phi,\nabla u\rangle_{L^2(\Lambda,\:\...
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0answers
160 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 ...
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146 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 ...
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44 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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1answer
108 views

“Designing” Nodal sets of Laplacians in 2 or 3 dimensional domains

The properties of nodal sets (i.e. zero level sets of eigenfunctions) for the first non-trivial eigenfunction for Laplacians have been studied extensively. My rough understanding is that one could ...
4
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1answer
249 views

heat kernel on closed manifolds - error in Chavel's book?

first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far. In Isaac Chavel's book Eigenvalues in Riemannian Geometry, ...
5
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1answer
183 views

certain smoothness of principal eigenvalue of Dirichlet Laplacian on polygons

For a given polygon $P_N$, with side lengths $x_1,\cdots,x_N$ and interior angles $\theta_1,\cdots,\theta_N$ let $\lambda(x_1,\cdots,x_N,\theta_1,\cdots,\theta_N)$ denote the least eigenvalue of ...
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1answer
199 views

Laplacian spectrum of directed network (digraph) and its complement

There is a well-known relation between the spectrum of graph laplacian and its complement's laplacian, namely $$λ_j (G^c) + λ_{n+2−j} (G) = n\;,$$ where the eigenvalues $λ_j$ are sorted in ...
2
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1answer
72 views

Universal constant for reverse inequality between first eigenvalues of Neumann and Dirichlet problems

I finally decided to post the following naive question but will if consensus is that it is out of the scope of this site , it will be immediately deleted. Suppose $\Omega\subset\mathbb R^2$ is a ...
2
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0answers
131 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 ...
5
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0answers
143 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(?) ...