Timeline for Poisson summation formula and its implication for the spectrum of the flat torus
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 30, 2019 at 8:27 | comment | added | Kostya_I | yes, I indeed messed up the LHS and RHS. | |
| May 30, 2019 at 8:24 | vote | accept | manifoldcurious | ||
| May 30, 2019 at 8:24 | comment | added | manifoldcurious | Again I apologise, I meant convergent when I wrote "true". Don't you mean that the RHS determines all $||\gamma||$? I think you might want to switch LHS and RHS in your latest comment. In that case I think I get it! I believe I needed the full argument, thank you! Anyway I will accept your answer. | |
| May 30, 2019 at 8:02 | comment | added | Kostya_I | Sorry, what do you mean by "both LHS and RSH series being true"? For each $t$, they are numerical expressions (associated to a particular torus), not statements. The argument goes as follows: suppose two tori are isospectral. Then, they have the same RHS. By Poisson identity, they also have the same LHS, for all $t$. But since LHS uniquely determines the collection of $||\gamma||$, they have the same length spectrum. This works the same way in the other direction. | |
| May 30, 2019 at 7:11 | comment | added | manifoldcurious | Sorry I was very sloppy in my explanation. Let $m_i$ be the multiplicity to $\lambda_1$. I was trying to say that I was expecting the following: for $r\in\mathbb{R}$ consider $E_r=e^{rt}(LHS-1)$. Then $r=\lambda_1$ is the unique value so that $E_{\lambda_1}\to m_1$ when $t\to\infty$. I was expecting $e^{rt}(RHS-1)$ to do something similar when $t\to\infty$. Another way I guess of stating my confusion is the following question: suppose I have both LHS and RHS series being true, but I do not have equality. Can I make the same conclusion about isospectrality? If not, why not? | |
| May 29, 2019 at 8:58 | comment | added | Kostya_I | I do not understand what the the equalities in your comment mean. Yes, both sides go to infinity as $t \to 0+$. The point is that the asymptotic expansion of the RHS allows you to recover $||\gamma||$ (with mupliplicities), as follows: the first term of the expasion is the prefactor of the RHS, the second one is the prefactor times $Ne^{-\alpha/4t}$, where $\alpha$ is the minimum $||\gamma||$ and $N$ its mupliplicities, etc. | |
| May 27, 2019 at 14:00 | comment | added | manifoldcurious | Sorry, this might be a stupid objection, but when $t\to 0+$ doesn't the LHS go to $\infty$? I would expect the argument to be something like $t\to 0 \implies LHS=eigenvalue=RHS=length$. Is the equality of the two series not preserved when taking the limit? | |
| May 25, 2019 at 9:19 | history | edited | Kostya_I | CC BY-SA 4.0 |
added 67 characters in body
|
| May 23, 2019 at 14:06 | history | answered | Kostya_I | CC BY-SA 4.0 |