Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 63 / 4 / d

Put $\lambda =2ks$. For $t\ge 0$, apply a valid Lieb-Robinson bound to each commutator in the supplied comparison of the two evolutions. For a disjoint shell at support distance $d$,In the usual uniform interaction-norm setting, truncation preserves the strength bound, so the same constants apply to $H_d$. The given polynomial shell-weight hypothesis and $e^{\lambda \tau }-1\le e^{\lambda t}$ now giveConsequentlyThe question explicitly supplies a bound for the full $H$; such a bound alone should not be assumed to transfer to $H_d$. A full-evolution comparison for truncated dynamics avoids that additional assumption. The Duhamel comparison formula givesNow use only the full-Hamiltonian bound, and sum the shell weights. Since ${\sum }_{q\ge d}{q}^{\alpha }{e}^{-\mu q}=O(d^{\alpha }{e}^{-\mu d})$, the errors for $j=d$ and $j=d-1$ are each bounded by $O(\Vert h_Z\Vert |t|d^{\alpha }{e}^{-\mu d+\lambda |t|})$. The triangle inequality comparing both evolutions to ${\tau }_t^H(h_Z)$ proves the same boxed estimate from the stated full-system assumption. Negative times follow by the reverse-time Duhamel formula; the commutator bounds use $|t|$.

For the interaction-term shells used to enforce $H_0=h_Z$, shell $d$ corresponds to support distance $d-1$ for distinct terms. Replace the displayed commutator estimate by the coarser bound with $e^{\lambda \tau }$, retaining the equal-time contribution for overlaps. Its distance factor is $e^{-\mu (d-1)}=e^{\mu }{e}^{-\mu d}$, and the fixed factor $e^{\mu }$ is absorbed by the big-O constant. This supplies the same claimed shell estimate without incorrectly using a zero equal-time bound for overlapping terms.