# Questions tagged [laplacian]

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

200 questions
Filter by
Sorted by
Tagged with
3k views

### Explicit eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary. Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known? An important example is the $n$-sphere ...
730 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 ...
1k 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 ...
1k views

522 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 $k$. ...
589 views

660 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 ...
216 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 $\Omega$....
353 views

4k 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 $\Delta$...