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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 $z=0$ …
Robert Bryant's user avatar
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. …
Robert Bryant's user avatar
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 $k\ge0$, …
Robert Bryant's user avatar
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 …
Robert Bryant's user avatar
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 …
Robert Bryant's user avatar
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 …
Robert Bryant's user avatar