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Results tagged with laplacian
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user 13972
The Laplacian matrix is the representation of a graph in matrix form.
4
votes
Accepted
Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?
Here is a sketch of an idea of how to show that the set $\mathcal{E}(g)\subset C^\infty(M)$ of all the eigenfunctions of the metric $g$ on a compact manifold $M$ determines $g$ up to a constant multip …
- 108k
22
votes
Accepted
Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order deriv...
As Raziel wrote, the local question is whether one can find a local basis of orthonormal vector fields that are divergence-free.
It's true that, in dimension $2$, this can only be done if the metri …
- 108k
12
votes
Accepted
Eigenfunctions restricted on closed geodesics
circle aren't usually eigenfunctions of the Laplacian on the circle. … of eigenspaces of the Laplacian on the great circle (I think it's about $\tfrac12(k{+}2)$ of them), but not into a single one of these eigenspaces. …
- 108k
8
votes
How to construct a scalar differential operator having the same spectrum as a non-scalar dif...
To get $\widehat{\Delta^1}$ to be the Bochner Laplacian, one must have $L = 0$, which is equivalent to $a''(u) = 2\bigl(a(u)-c\bigr)$. … Thus, there is a nontrivial $2$-parameter family of metrics on the $2$-sphere such that the Bochner Laplacian has the desired intertwining property. …