Timeline for Asymptotic question about time ordered exponentials
Current License: CC BY-SA 3.0
9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Dec 9, 2016 at 6:53 | comment | added | lcv | I guess that case can be somehow reduced to the previous one factoring out the growth bound of $A(t)$. | |
| Dec 9, 2016 at 6:52 | comment | added | lcv | Provided some smootheness of $A(t)$ and a no level crossing $B$ can be expanded into a series in $1/r$. One of the best reference on this is still Kato's classic journals.jps.jp/doi/abs/10.1143/JPSJ.5.435. The adiabatic theorem has been generalized to the case where $A(t)$ is a generator of a contraction semigroup for all $t$. This still implies nicely bounded solution and an adiabatic theorem in the same vein. You seem to be interested to a case where $A(t)$ can have eigenvalues with positive real part. | |
| Dec 9, 2016 at 6:45 | comment | added | lcv | It's maybe too late to comment on this but your question is related (except for one important point) to the adiabatic theorem of quantum mechanics. You are looking for an expansion of $B$ for large $r$ where $B$ is solution of $(1/r)\dot{B} = A(t) B(t)$ (dot indicates differentiation with respect to time). This is also called singular perturbation theory because when $r=\infty$ formally the character of the ODE changes. In quantum mechanics $A(t)$ is antihermitian for all $t$ meaning its eigenvalues are purely imaginary. This in turn implies that $B$ is unitary. | |
| Apr 30, 2012 at 23:05 | answer | added | David E Speyer | timeline score: 3 | |
| Apr 27, 2012 at 18:59 | vote | accept | David E Speyer | ||
| Feb 16, 2012 at 8:48 | history | edited | user2035 |
tag typo
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| Dec 1, 2011 at 9:40 | answer | added | Jon | timeline score: 7 | |
| Nov 30, 2011 at 13:09 | history | asked | David E Speyer | CC BY-SA 3.0 |