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Results tagged with sp.spectral-theory
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user 13972
Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
12
votes
Accepted
Eigenfunctions restricted on closed geodesics
The answer is 'no', as you can see by taking the case of $M$ being the unit $2$-sphere in $\mathbb{R}^3$ and the geodesic $\gamma$ being a great circle, say, the horizontal great circle given by $z=0$ …
4
votes
Accepted
Diagonalization of a matrix of differential operators
Another approach, for this particular example is to try to solve the equation $AMA^{-1} - m I = 0$, where $A$ is an invertible $2$-by-$2$ matrix of functions and $m$ is a scalar differential operator. …
23
votes
Accepted
Eigenfunctions of the laplacian on $\mathbb{CP}^n$
The $k$-th eigenfunctions are actually easy to describe: In $\mathbb{C}^{n+1}$ with unitary complex coordinates $z_0,z_1,\ldots,z_n$, write $Z = |z_0|^2+\cdots+|z_n|^2$.
Now, for a given $k\ge0$, …
17
votes
Accepted
What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?
Of course, Igor's answer points the way to working out the answer the OP wanted, but it may not be clear, even after you have got the eigenvalues, what the corresponding eigenfunctions are, or that th …
8
votes
How to construct a scalar differential operator having the same spectrum as a non-scalar dif...
I thought a little bit about your question, which is phrased a little more generally than I like, but I decided to think about it with the restrictions that $\widehat{\Delta^0}$ be a differential oper …
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 …
11
votes
Accepted
On eigenfunctions of the Laplace Beltrami operator
For $\mathrm{SU}(2)$, with the scale for the biïnvariant metric so that it becomes isometric to the unit $3$-sphere $S^3$ in Euclidean $4$-space, it is well-known what the eigenvalues of the Laplace-B …