Timeline for Dimension of spectral projection subspaces under local convergence
Current License: CC BY-SA 4.0
9 events
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| Feb 13 at 21:45 | comment | added | Christian Remling | @Keen-ameteur: I think I have a slightly clearer picture now, and it seems this is still not working, though there are similar in spirit statements that are true. See the latest edit please. | |
| Feb 13 at 21:43 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Feb 13 at 16:35 | comment | added | Keen-ameteur | I think I may have misstated the condition I wanted. I've edited the question. Under my original intention and my corrected edit, $q_n\leq n-1$ for your proposed $V_n$. I'm sorry for the confusion. | |
| Feb 13 at 15:35 | comment | added | Christian Remling | @Keen-ameteur: Sorry, it's not completely clear to me what you are saying in your last comment, but I edited my answer and I hope this will address everything. Also, $PH_nP$ is not the zero operator in my example for a fixed projection, it's the (restricted) Laplacian. | |
| Feb 13 at 15:35 | history | edited | Christian Remling | CC BY-SA 4.0 |
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| Feb 13 at 15:19 | comment | added | Keen-ameteur | It is very likely that I am confused, but $H_n$ have to stabilize near the origin. Namely, if $H_n\to 0$ then $M_n\equiv 0$. I think that for all $n\geq n_0$, we have $P_{q_{n_0}}H_nP_{q_{n_0}} = M_{n_0}$. So if $H_n\to 0$ that must mean that $M_{n_0}\equiv 0$, doesn't it? | |
| Feb 13 at 15:06 | comment | added | Christian Remling | @Keen-ameteur: This doesn't change anything essentially. I only need to take $q_n$ sufficiently large and will then still have an eigenvalue $\simeq E$. Admittedly, it will have moved a bit, but we then have $\dim\chi_{(E-\epsilon,E+\epsilon)}(M_n)\ge 1$, $(E-\epsilon,E+\epsilon)\cap\sigma (\Delta)=\emptyset$. | |
| Feb 13 at 14:58 | comment | added | Keen-ameteur | Thanks for your answer. I was asking that $H_n$ projected onto the finite box, what I denoted by $M_n$, have $E$ in its spectrum with multiplicity bounded from below. It seems like in the example you wrote, $\dim \chi_{\{ E\}}(M_n)= 0$ eventually. | |
| Feb 13 at 14:32 | history | answered | Christian Remling | CC BY-SA 4.0 |