Timeline for Morse index in PDEs
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 17, 2022 at 14:14 | vote | accept | Jingeon An-Lacroix | ||
| Aug 16, 2022 at 15:11 | answer | added | Otis Chodosh | timeline score: 10 | |
| Aug 16, 2022 at 8:43 | answer | added | Thomas Rot | timeline score: 3 | |
| Aug 9, 2022 at 19:47 | comment | added | Jingeon An-Lacroix | @mme Okay, thank you again for the comment :) | |
| Aug 9, 2022 at 18:05 | comment | added | mme | I am not familiar with the context, so I can't comment in any real detail. Whatever the geometric interpretation of "Morse index" is --- if it exists --- will depend on that context, and may be difficult to prove. To my eye, the main theorem of [1] is that there are in fact geometric consequences of the analytic assumption of finite Morse index. If you want to get some intuition for the idea of Morse index appearing in [2], I would encourage you to see how it is used (and therefore why it is relevant) in that paper. | |
| Aug 9, 2022 at 16:09 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor formatting and additions
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| Aug 9, 2022 at 7:21 | comment | added | Jingeon An-Lacroix | @mme Thank you for the comment. Now I see the definition above should have inspired by the one in the differential topology. But I am still not sure about the motivation of definition. In the literature [2], $L_+$ is given by $L_+:=(-\Delta)^s+V$ where $s\in(0,1)$ and some potential $V$. Does "Morse index" defined above says something geometrical property about its eigenfunctions? | |
| Aug 8, 2022 at 22:31 | comment | added | mme | In uses of the term "Morse index" I know in differential topology, it refers to the number of negative eigenvalues of Hess_x f, where x is a critical point of the Morse function f. This is consistent with what you see elsewhere. | |
| Aug 8, 2022 at 22:14 | history | asked | Jingeon An-Lacroix | CC BY-SA 4.0 |