Timeline for Local energy estimate in a semiclassical regime
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 31, 2022 at 6:52 | comment | added | J.Mayol | Looks like a change of variable could be suffient: doing $x \mapsto x+1$ then we need to localize the new $x$ as $x = \mathcal{O}(\varepsilon)$, so introducing $x=\varepsilon \tilde{x}$ we obtain $(\varepsilon^{-3}h^2D_{\tilde{x}}^2 + \tilde x)u=0$ (changing charts, etc.). But for $\varepsilon= Ch^{2/3}$ we have $\varepsilon^{-3}h^2$ which is not a semiclassical parameter anymore ... | |
| May 26, 2022 at 7:00 | comment | added | J.Mayol | I edited so that we only consider the $h_n$ values. What I want is the $O(\varepsilon^{1/4})$ bound that I'm asking, which is an upper bound (of course if you really look at the precise asymptotics of Hermite functions you will see that in this special case this is also a lower bound, but here I only want the upper bound. | |
| May 26, 2022 at 2:29 | comment | added | Willie Wong | My (somewhat facetious) point is that there does exist a much simpler proof, albeit not the one you are looking for. So as the question asker, it may help if you specify more precisely what is the "lower bound" of simplicity you are looking for. | |
| May 25, 2022 at 19:02 | history | edited | J.Mayol | CC BY-SA 4.0 |
added 80 characters in body
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| May 25, 2022 at 19:01 | comment | added | J.Mayol | @WillieWong the thing is to not use what is known about the Hermite polynomials. In fact, because it is semiclassical analysis one can change $x^2$ for some $V(x)$ with similar properties. For the fact that $Pu_h$ is not always solvable, just take $h=h_n=\frac{1}{\sqrt{2n+1}}$ (note that there is a square root missing in your $h$, it's because you should also rescale de $x^2$). | |
| May 24, 2022 at 20:11 | history | edited | J.Mayol | CC BY-SA 4.0 |
added 91 characters in body
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| May 24, 2022 at 20:10 | comment | added | J.Mayol | ahh right, I'm assuming that, so I will edit and add it, thanks | |
| May 24, 2022 at 18:30 | history | asked | J.Mayol | CC BY-SA 4.0 |