Timeline for Asymptotic question about time ordered exponentials
Current License: CC BY-SA 3.0
8 events
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| Jul 2, 2013 at 16:05 | history | wiki removed | Robert Cartaino | ||
| May 2, 2012 at 9:35 | comment | added | Jon | David, I cannot do statements about combined solutions of more differential equations but I can show you how precise is my approximation for the first example you gave. I solved numerically the equation amd compared with the approximate solution. The agreement is strikingly good. | |
| May 1, 2012 at 20:25 | history | edited | David E Speyer | CC BY-SA 3.0 |
added 1931 characters in body
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| May 1, 2012 at 13:23 | comment | added | Jon | Fine. Give me a few time to work out completely this example and expand my answer. For this I have completely evaluated eigenvectors and eigenvalues at leading order. It is new also for me as I have always applied this to quantum mechanics. | |
| May 1, 2012 at 12:02 | comment | added | David E Speyer | Right, they are not Hermitian. As stated in the original question, they obey $A(-t) = A(t)^{\ast}$. This has the effect that the total transport along the curve is Hermitian, but it is made up out of a lot of non Hermitian things. | |
| May 1, 2012 at 10:28 | comment | added | Jon | Ok, I have found at least a couple of problems with your example. I think that at the foundation of your confusion lies the fact that you are not working with Hermitian self-adjoint matrices. This has the important implication that you must have left and right eigenvector, let us say $v_{\pm}$ and $u_{\pm}$ and so, the series takes eventually the form $$e^{kr}v_+^Tu_++e^{-kr}v_-^Tu_-+\ldots$$. Finally, you are systematically omitting the geometric contribution going like $\exp{\pm\int_0^tdt'v_\pm^T\frac{d}{dt'}u_\pm}$ and this cannot be done here. Do you need an explicit example? | |
| May 1, 2012 at 8:54 | comment | added | Jon | David, you missed the geometric phases here. These terms go like $e^{\int_0^t dt'u_1\dot u_1^T}\, e^{\int_0^t dt'v_1\dot v_1^T}$ and should be included. I will take some time to work out this example. | |
| Apr 30, 2012 at 23:05 | history | answered | David E Speyer | CC BY-SA 3.0 |