The laplacian tag has no usage guidance.

**16**

votes

**1**answer

360 views

### Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...

**14**

votes

**3**answers

595 views

### Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators

EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...

**14**

votes

**1**answer

386 views

### Weyl's law on asymptotic of Laplacian vs Hilbert's theorem on degree of a projective variety

Hi,
Let $M$ be a compact Riemannian manifold of dimension $n$. Define
the integer-valued function $N(k)$ to be the number of eigenvalues of the Laplacian on $M$
which are less than or equal to $k$. ...

**13**

votes

**4**answers

1k views

### High multiplicity eigenvalue implies symmetry?

It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions ...

**10**

votes

**2**answers

239 views

### First eigenvalue of the Laplacian on a regular polygon

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:
...

**10**

votes

**1**answer

276 views

### heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...

**9**

votes

**3**answers

1k views

### The first eigenvalue of the laplacian for complex projective space

What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?

**9**

votes

**3**answers

381 views

### Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph,
One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...

**9**

votes

**2**answers

336 views

### Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$.
On the other ...

**8**

votes

**3**answers

308 views

### Spectrum of Dirichlet Laplacian on a Parallelogram

Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and ...

**8**

votes

**2**answers

803 views

### Characterize where the Dirichlet Problem for the Laplacian is always solvable

Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...

**7**

votes

**1**answer

254 views

### Eigenfunctions restricted on closed geodesics

Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l_1\mathbb{Z}\oplus l_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial ...

**7**

votes

**2**answers

295 views

### Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold

It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the ...

**7**

votes

**3**answers

325 views

### Criteria for Positivity of Pseudoddifferential Operators on Manifolds

Let $(M,g)$ be a Riemannian Manifold and $L^2$ the Hilbert space given by the volume form associated to the metric. Let $L_0^2$ be the subspace which is orthogonal to the constant functions. When is ...

**7**

votes

**2**answers

605 views

### Adjoint of a Connection Using the Hodge Map?

For a Riemannian manifold $(M,g)$ with exterior derivative d, the codifferential d$^\ast$ is defined to be the unique map for which
$$
g(\omega,d\omega') = g(d^* \omega,\omega'), ~~~ \omega,\omega' ...

**7**

votes

**2**answers

1k views

### Eigenvectors and partitions of graphs

Let G be an undirected graph with the node set $V$ and the Laplacian matrix $L$. Let $N(v)$ denote the neighbors of a node $v$ and $|N(v)|$ its degree. Then a partition $\pi=(V_1, V_2, \ldots, V_k)$ ...

**7**

votes

**1**answer

923 views

### First eigenvalue of the Laplacian on Berger spheres

Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the ...

**6**

votes

**1**answer

420 views

### First nonzero eigenvalue of the Laplacian on the submanifold

Consider a compact, connected $n$ dimensional Riemmanian manifold $\mathcal{N}$ and its $m$ dimensional closed submanifold $\mathcal{M}$ (with the metric coming from from the one defined on ...

**6**

votes

**1**answer

262 views

### Fractional Laplacian and stereographic projection

The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere ...

**6**

votes

**1**answer

78 views

### Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...

**6**

votes

**1**answer

268 views

### Laplace-Beltrami operator on a Lie group

For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...

**5**

votes

**1**answer

778 views

### Growth of Laplacian eigenvalues on a compact domain?

Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct ...

**5**

votes

**1**answer

857 views

### How to construct a scalar differential operator having the same spectrum as a non-scalar differential operator exploiting symmetries?

I am interested in eigenvalue problems for differential operators acting on one forms on closed two-dimensional manifolds and how they relate to eigenvalue problems of associated operators acting on ...

**5**

votes

**1**answer

477 views

### laplace equation on manifolds with boundary

in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f $ if ...

**5**

votes

**2**answers

1k views

### Eigenvalues of Laplacian

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be
$$ \#\{v < A^2\} = ...

**5**

votes

**1**answer

605 views

### Multiplicity of eigenvalues of the Laplacian on quaternionic projective space

Using the classic spherical harmonics theory, one obtains the $k$-th eigenvalue of the $n$-dimensional round sphere $S^n$ to be $k(k+n-1)$, and its multiplicity is $\binom{n+k}{k}-\binom{n+k-1}{k-1}$, ...

**5**

votes

**2**answers

210 views

### Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to
$$
-\Delta\psi = ...

**5**

votes

**1**answer

614 views

### Courant nodal domain Theorem for sums of eigenfunctions?

Courant's nodal domain theorem gives a bound on the number of nodal domains for an eigenfunction of the Laplacian. Namely, if $M$ is a smooth compact Riemannian manifold, and $f$ is an eigenfunction ...

**5**

votes

**1**answer

370 views

### About the quantum spectrum of a certain potential.

Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ is compact (..is there a weaker criteria?..) then the spectrum ...

**5**

votes

**0**answers

294 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( ...

**4**

votes

**2**answers

411 views

### Closed formula for heat kernel

Is there, similar to the Mehler kernel, a closed formula for the heat kernel of the heat equation associated to the Laplacian
$$ -\sum_j \frac{d^2}{dx_j^2} + 2\sqrt{-1} \sum_j \lambda_j \frac{d}{dx_j} ...

**4**

votes

**4**answers

1k views

### Boundaries of the eigenvalues of a symmetric matrix (or of its Lapacian)

Given the adjacency matrix $A_{ij}$ of a graph with $N$ vertices and $M$ links (or any binary symmetric matrix of size $N \times N$), is it possible to establish lower and upper boundaries of its ...

**4**

votes

**2**answers

451 views

### What is the Weitzenböck formula for the $\bar\partial$-Laplacian

It is well-known that the Weitzenböck formula for the real Laplacian is
$$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$
where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the ...

**4**

votes

**2**answers

335 views

### How does a quasi-isometry affect Poisson or Martin boundaries?

Take two graphs (of bounded valency) or manifolds (f bounded geometry) $G$ and $G'$. Assume there is a quasi-isometry $f:G \to G'$, and assume the Poisson or Martin boundary of $G$ is known, what may ...

**4**

votes

**1**answer

180 views

### Divisibility Relation for Spanning Trees of a Graph

Let $A = \big[{1\ 1\atop 1\ 0}\big]$, and let $G_n$ be the graph whose adjacency matrix is
$A^{\otimes n}$. Also let $\kappa(G)$ denote the number of spanning trees of $G$. From a significant amount ...

**4**

votes

**1**answer

152 views

### Laplace-Beltrami and averaging

For a Riemannian manifold $M$ with metric $g$ and Laplace-Beltrami operator $-\Delta_{g}$, what conditions on $M$ guarantee that $-\Delta_{g} u(x)$ measures the difference between $u(x)$ and the ...

**3**

votes

**2**answers

583 views

### Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...

**3**

votes

**1**answer

489 views

### Relation between the de Rham and Hodge Laplacians on the Exterior Algebra

For a Kahler manifold $M$ we have two well-known Laplacians: the de Rham Laplacian $\Delta_{\text{d}} = ($d$ + $d$^\ast)^2$, and the Dolbeault Laplacian $\Delta_{\overline{\partial}} = ...

**3**

votes

**2**answers

692 views

### Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature

A friend in physics asked this question, and I didn't know the answer.
Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...

**3**

votes

**3**answers

2k views

### Integral kernel for the resolvent of the laplace operator

Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of ...

**3**

votes

**1**answer

136 views

### Gagliardo Nirenberg inequality for the laplacian

It is a classical result due to Gagliardo and Nirenberg that there exists a constant C such that it holds
$$
||\nabla \psi|| _{L ^\infty (\mathbb{R}^2)} ^2
\le ||D ^2 \psi|| _{L ^\infty ...

**3**

votes

**3**answers

169 views

### Limits for eigenvalues for the Dirichlet Laplacian

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$
\begin{cases}
-\Delta u=\lambda u & \mbox{in }\Omega\\
u=0 & \mbox{on ...

**3**

votes

**1**answer

313 views

### Calculating a generalized inverse (Moore–Penrose pseudoinverse)

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix:
\begin{equation}
A_n=
\begin{pmatrix}
0 & 0 & 0 &\cdots & 0 & 0 & 1\\
0 & 0 ...

**3**

votes

**1**answer

257 views

### Graph Laplacian simple eigenvalues

Is there a class of graphs (besides the path graphs) for which we know that the Laplacian L = D - A (where D is the degree matrix and A is the adjacency matrix) has simple spectrum, i.e. all Laplacian ...

**3**

votes

**2**answers

153 views

### Boundedness of Solutions to $\Delta u = f u$ on $R^2$

Consider the Laplacian $\Delta = d/dx^2 + d/dy^2$ on $\mathbb{R}^2$.
This is true: Let $f$ be a nonnegative function, not identically zero. Then any positive solution of $\Delta u = f u$ is ...

**3**

votes

**1**answer

184 views

### The complex heat kernel on a Riemann manifold

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial ...

**3**

votes

**0**answers

130 views

### On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation
$$\triangle \varphi = 4 \pi \rho \qquad (1)$$
defined on 3-dimensional oriented Riemannian manifold $(M,g)$,
where $g$ is metric and ...

**3**

votes

**0**answers

117 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 ...

**2**

votes

**1**answer

452 views

### The smallest positive eigenvalue and the length of the shortest geodesic

I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ ...

**2**

votes

**3**answers

869 views

### How to define Laplacian on $L_2$

This might be a dumb question, but I thought the Laplacian (classical) is defined for $C^2$ functions. How do we extend that to be a self-adjoint operator on all of $L_2$?
Is it the so called ...