Unresolvable Identifier

arXiv:2604.18506 · detector doi_compliance · cross_source · 2026-05-20 03:59:26.711503+00:00

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M.DFeit,J.AFleck,andASteiger.“SolutionoftheSchrödinger equationbyaspectralmethod”.In:JournalofComputational Physics47.3(1982),pp.412–433.issn:0021-9991.doi:https: //doi.org/10.1016/0021-9991(82)90091-2.url:https://www. sciencedirect.com/science/article/pii/0021999182900912. Appendix A. Magnus expansion TheMagnusexpansionprovidesaconvenientframeworkforcomput- ingthetimeevolutionofaquantumstatefromoneinstanttoanother byexpressingthetime-evolutionoperatorastheexponentialofan effectiveoperator,denotedby ΩΩΩ. Thisoperatoradmitsaseriesrepre- sentation,wheretheresultingapproximationorderisdeterminedby thetruncationorder 𝑝considered. Afterpartitioningthetimeaxis into𝑛𝑤windows,anoperator ΩΩΩ(𝑤)canbecomputedforeachwindow, which propagates the state from the beginning to the end of that window, as expressed in(2.12). Takingℋℋℋ tot(𝑡)as the Hamiltonian governing the dynamics, the first three discrete orders ofΩΩΩ(𝑤)are reportedbelow. ΩΩΩ(𝑤) 1 = −𝑖∆𝑡 𝑚−1∑ 𝑗=0 ℋℋℋ tot ( 𝑡(𝑤) 𝑗 ) , ΩΩΩ(𝑤) 2 = −∆𝑡2 2 𝑚−1∑ 𝑗1=0 𝑗1−1∑ 𝑗2=0 [ ℋℋℋ tot ( 𝑡(𝑤) 𝑗1 ) ,ℋℋℋ tot ( 𝑡(𝑤) 𝑗2 )] , ΩΩΩ(𝑤) 3 = 𝑖∆𝑡3 6 𝑚−1∑ 𝑗1=0 𝑗1−1∑ 𝑗2=0 𝑗2−1∑ 𝑗3=0 ([ ℋℋℋ tot ( 𝑡(𝑤) 𝑗1 ) , [ ℋℋℋ tot ( 𝑡(𝑤) 𝑗2 ) ,ℋℋℋ tot ( 𝑡(𝑤) 𝑗3 )]] + [ ℋℋℋ tot ( 𝑡(𝑤) 𝑗3 ) , [ ℋℋℋ tot ( 𝑡(𝑤) 𝑗2 ) ,ℋℋℋ tot ( 𝑡(𝑤) 𝑗1 )]]) . (A.1) Theerrorinthetimeevolutioncausedbytruncatingtheexpansion can be derived as follows. Let𝒰𝒰𝒰(𝑇) =∏ 𝑛𝑤 𝑤=1𝒰𝒰𝒰(𝑤)be the exact evo- lution split into𝑛𝑤windows of widthℎ=𝑇∕𝑛𝑤, where each exact windowpropagatoris 𝒰𝒰𝒰(𝑤)=𝒯exp (−𝑖∫ 𝑡(𝑤) 𝑡(𝑤−1

Evidence payload

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