Timeline for Smooth perturbation of a positive self-adjoint operator with compact resolvent
Current License: CC BY-SA 3.0
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| Feb 23, 2020 at 17:41 | comment | added | marmistrz | The link to the paper by Jerry Kazdan is dead, here's a working one: web.archive.org/web/20200223174046/https://www.math.upenn.edu/… | |
| Nov 20, 2015 at 18:47 | comment | added | Willie Wong | The author gave 3 references to that claim. Do none of them answer you question? @HSM | |
| Nov 20, 2015 at 17:57 | comment | added | SMS | @WillieWong Nice comment, that is what I was sure is correct. However, I just found this paper: arxiv.org/pdf/0710.3947.pdf. Please take a look at Remark 1.2 on page 2. If the perturbation in question is Ricci flow, how is the author guaranteeing that the eigenfunctions are $C^1$? I am a bit confused now. I don't see that the Ricci flow gives any additional handle on whether the eigenvalues are distinct. | |
| Nov 17, 2015 at 18:37 | comment | added | Peter Michor | The papers cited in mathoverflow.net/a/108630/26935 give a quite complete answer to your question. In particular, see the beginning of mat.univie.ac.at/~michor/DC-perturb.pdf. | |
| Nov 17, 2015 at 18:21 | review | Close votes | |||
| Nov 18, 2015 at 20:03 | |||||
| Nov 17, 2015 at 18:12 | comment | added | Willie Wong | See also: mathoverflow.net/questions/43124/… and math.upenn.edu/~kazdan/509S07/eigenv5b.pdf | |
| Nov 17, 2015 at 18:09 | comment | added | Willie Wong | Something nasty does happen when eigenvalues have multiplicities. I believe these are all discussed already in Kato's book. Summary: when eigenvalues are distinct the smoothness of the eigenprojections correspond to that of the parameter. When they are not distinct the eigenvectors need not even be continuous. (You can even see this for symmetric $2\times2$ matrices.) | |
| Nov 17, 2015 at 17:29 | history | edited | user82891 |
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| Nov 17, 2015 at 16:33 | history | edited | user82891 | CC BY-SA 3.0 |
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| Nov 17, 2015 at 16:25 | review | First posts | |||
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| Nov 17, 2015 at 16:23 | history | asked | user82891 | CC BY-SA 3.0 |