Timeline for Does a spectral gap lift to covering spaces?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 17, 2018 at 11:28 | comment | added | Jan Bohr | I have found the following result: $\vert \mathrm{tr}(e^{-t\Delta_M})- \dim \ker \Delta_M \vert \le C e^{-t \lambda(M)}$. But it only seems to hold if $\Delta_M$ is discrete. | |
| Jul 17, 2018 at 10:53 | comment | added | Jan Bohr | No, $0$ is allowed. Note that I've defined $\lambda(M)$ as the infimum of all nonzero spectral points. I think one can ask a related question, namely: If $0\notin \sigma(\Delta_N)$, do we also have $0\notin \sigma(\Delta_{\hat N})$? However I'm also interested in the general case, where $0\in \sigma(\Delta_N)$, but it is isolated in the spectrum. | |
| Jul 17, 2018 at 10:42 | comment | added | R W | Do you exclude the 0 eigenvalue for compact manifolds.? | |
| Jul 17, 2018 at 10:36 | comment | added | Jan Bohr | In the linked math.stackexchange question I calculate that $\lambda(S^1_L) = (2\pi/L)^2$ (Where $S^1_L$ is a circle with length $L$). Doesn't this give rise to a counterexample to your claim that finite covers have the same spectral gap? | |
| Jul 17, 2018 at 10:29 | comment | added | Jan Bohr | Can you suggest a reference that discusses the interpretation of the spectral gap in terms of exponential decay rate of the heat kernel? Of course preferably on non-compact manifolds. | |
| Jul 17, 2018 at 10:19 | history | answered | R W | CC BY-SA 4.0 |