hamburgminicourse2017:time-dependent_potentials
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| hamburgminicourse2017:time-dependent_potentials [2017/04/02 22:04] – markus | hamburgminicourse2017:time-dependent_potentials [2017/04/02 22:06] (current) – markus | ||
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| We used the fact that the Hamitonians commute with the short-stime evolution operators at the same time-step in this. | We used the fact that the Hamitonians commute with the short-stime evolution operators at the same time-step in this. | ||
| - | 5. Now we rewrite the $H(t)H(s)^{-1}$ encounters as $K(t,s) + \mathrm{id} : \mathcal{H} \rightarrow \mathcal{H}$ where the operator $K(t,s)$ is bounded by $L |t-s|$, $L$ the Lipschitz constant of $V$ in the Sobolev--Kato--Lipschitz space. We have now: | + | 5. Now we rewrite the $H(t)H(s)^{-1}$ encounters as $K(t,s) + \mathrm{id} : \mathcal{H} \rightarrow \mathcal{H}$ where the operator $K(t,s)$ is bounded by $L |t-s|$ |
| \[ | \[ | ||
| U_k(t,0) = H(t_{k-1})^{-1} U_k^{(k)} \prod_{j=1}^{k-1} (K(t_j, | U_k(t,0) = H(t_{k-1})^{-1} U_k^{(k)} \prod_{j=1}^{k-1} (K(t_j, | ||
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| \end{align*} | \end{align*} | ||
| This convergence is uniform because $t$ is from a bounded set $[0,T]$. Finally already (6) showed that the evolution operators are uniformly bounded on $D(H)$ and thus can extended to the whole Hilbert space. | This convergence is uniform because $t$ is from a bounded set $[0,T]$. Finally already (6) showed that the evolution operators are uniformly bounded on $D(H)$ and thus can extended to the whole Hilbert space. | ||
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hamburgminicourse2017/time-dependent_potentials.1491163493.txt.gz · Last modified: 2017/04/02 22:04 by markus