Timeline for Can you compute one eigenspace without computing them all?
Current License: CC BY-SA 4.0
14 events
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| Aug 26, 2021 at 15:27 | comment | added | Leo Moos | @paulgarrett Right, that's maybe a better way of putting it. Obviously I've done a poor job of communicating my intent, but I hope it's a bit clearer now. | |
| Aug 26, 2021 at 15:16 | comment | added | paul garrett | Ah, well, I don't know an approach that gets the low ones "easily" and not all... | |
| Aug 26, 2021 at 15:11 | comment | added | Leo Moos | @paulgarrett For sure, although I should say that I've reformulated my question because I'm worried that it's been misunderstood. Above all I am looking for an argument that allows for a computation of the 'low' eigenvalues without involving the 'higher' ones in the argument. (I mean not only their numerical values, but steps that would involve a result like the lemma I cite in the question would also be forbidden.) I'd be happy to accept an answer that applies only to the sphere, provided it conforms to this restriction. | |
| Aug 26, 2021 at 15:00 | history | edited | Leo Moos | CC BY-SA 4.0 |
reformulated question
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| Aug 26, 2021 at 14:54 | comment | added | paul garrett | As @RomainGicquaud comments, the sphere example is completely and definitively answered by a little representation theory... Would such an answer be helpful to you, even if somewhat contrary to your expectations? | |
| Aug 26, 2021 at 13:35 | comment | added | Romain Gicquaud | It also gives a reference to the book "Harmonic analysis on homogeneous spaces" by Nolan Wallach where you will find more details. As quoted in an answer below, the Laplacian is (up to some constant) the Casimir operator of the representation. | |
| Aug 26, 2021 at 13:15 | comment | added | Leo Moos | @RomainGicquaud I don't know enough about representation theory to tell whether this answers the question, although I'm a bit skeptical because Renato Bettiol's answer there points to the same reference I quote here. | |
| Aug 26, 2021 at 13:00 | comment | added | Romain Gicquaud | For spheres, tori and more generally for (compact) homogeneous spaces, you can rely on representation theory by the fact that the Laplacian commutes with the action of the symmetry group. This is how the spectrum of the sphere (and e.g. the complex projective space) are computed, see e.g. mathoverflow.net/questions/85481/… | |
| Aug 26, 2021 at 12:56 | comment | added | Leo Moos | @RomainGicquaud I've modified some modifications that will hopefully clarify the question. I'm not asking about an argument applicable to an arbitrary manifold, but specifically to the round sphere and/or products. | |
| Aug 26, 2021 at 12:54 | history | edited | Leo Moos | CC BY-SA 4.0 |
clarified question; fixed typos
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| Aug 26, 2021 at 12:41 | comment | added | Romain Gicquaud | So you mean you want to find the exact first eigenvalues of some manifold? You should be more precise regarding the manifolds you are considering. I don't think that there is a procedure to find the eigenvalues in general. Rayleigh quotient is useful to get estimates. | |
| Aug 26, 2021 at 12:35 | comment | added | Leo Moos | @RomainGicquaud While this is true, I'm not sure it's applicable here. Do you know of a way to use the Rayleigh quotients to calculate the spherical harmonics, for example? | |
| Aug 26, 2021 at 12:23 | comment | added | Romain Gicquaud | Normally, eigenvalues and eigenspaces are computed using a min-max argument with the Rayleigh quotient. This gives you one eigenvalue at a time. | |
| Aug 26, 2021 at 11:07 | history | asked | Leo Moos | CC BY-SA 4.0 |