Timeline for Weyl law for (non-semiclassical) Schrodinger operator
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 17, 2019 at 9:50 | answer | added | Victor Ivrii | timeline score: 4 | |
| Apr 16, 2019 at 16:54 | comment | added | Hadrian Quan | Apologies for not realizing you meant this. I think you're right, as the principal symbol in classical calculus he considers will not include the potential $V(x)$ (I suppose this is the point of considering semiclassical operators!) I am glad you were able to find some references. | |
| Apr 16, 2019 at 15:58 | answer | added | Maxim Braverman | timeline score: 4 | |
| Apr 16, 2019 at 15:21 | comment | added | Maxim Braverman | @HadrianQuan Clearly, I assume that $V(x)$ grows at infinity, so that the volume in (**) is finite. I did look at Hormander. I don't think the result I need follows from it. | |
| Apr 14, 2019 at 17:24 | comment | added | Hadrian Quan | I don't think completeness alone will suffice to say anything this general. E.g. there are complete riemannian manifolds with purely continuous spectrum (such as the upper half-plane $\mathbb{H}^2$ has spectrum $[1/4,\infty)$.) If you're not considering closed manifolds, and you don't assume something about how your potential $V(x)$ behaves at infinity I don't think you can hope to get a Weyl Law. You might look at H\"{o}rmander's "spectral function of an elliptic operator" since he proves a weyl law very generally, though at the cost of some FIO theory being necessary. | |
| Apr 14, 2019 at 6:44 | comment | added | Josiah Park | Victor Ivrii is somewhat responsive to emails. One could try emailing the question to him for a response. | |
| Apr 13, 2019 at 15:52 | history | edited | Maxim Braverman | CC BY-SA 4.0 |
added 11 characters in body
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| Apr 12, 2019 at 18:42 | history | asked | Maxim Braverman | CC BY-SA 4.0 |