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Extra resources for Computational Methods for Large Systems
82) ret † so that R,L = −2 R,L . 81) is the desired relation. 81) is most useful whenever (1) one can give recursive algorithms, so can be calculated without having to deal with the full Hilbert space at a time, or (2) one can design approximations for so that it is not necessary to deal with the Hilbert space of the leads at all. , comprises a sufficiently large part of the leads). Almost all scientific works that perform a channel decomposition begin by rewriting Eq. 83) so that by construction, T (E) = TrC ττ† .
6) exists, at least in principle, which allows reconstruction of the probing potential φex (t) from knowledge of n(t) (and i ) at all times t ≥ 0. 1 Runge–Gross Theorem The Runge–Gross theorem emphasizes that the time evolution of the density n(t) is a unique characteristic of the probing potential φex (t): Two probing fields, which differ by more than a homogeneous shift in space, invoke two different density evolutions. This insight is then later used to argue that a density profile, n(rt), that is driven in one system with interaction Uˆ by φex (t) can also be seen in another system with a different interaction Uˆ after φex (t) has been replaced by the appropriate modulation φex (t).
Almost all scientific works that perform a channel decomposition begin by rewriting Eq. 83) so that by construction, T (E) = TrC ττ† . Authors interpret τ as a transmission matrix and hence identify the eigenvectors of ττ† as the transmission channels. We wish to point out here that this widespread practice has to be taken with a grain of salt. 1. The trace in Eq. 81) is over the states of the central region and not over the (transverse) Hilbert space of the leads. Ironically, this is why we have derived it in the first place.