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# Review of 'Nonsmooth and level-resolved dynamics illustrated with the tight binding model'

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A very interesting work on first-order time-dependent perturbation theory; more clarifications help.
 Average rating:     Rated 3.5 of 5. Level of importance:     Rated 3 of 5. Level of validity:     Rated 3 of 5. Level of completeness:     Rated 3 of 5. Level of comprehensibility:     Rated 5 of 5. Competing interests: None

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### Nonsmooth and level-resolved dynamics illustrated with the tight binding model

(2014)
We point out that in the first order time-dependent perturbation theory, the transition probability may behave nonsmoothly in time and have kinks periodically. Moreover, the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is a simple equality about the sinc function $$\sinc x \equiv \sin x / x$$. These physical effects appear in many systems with approximately equally spaced spectra, and is also robust for larger-amplitude coupling beyond the domain of perturbation theory. We use a one-dimensional periodically driven tight-binding model to illustrate these effects, both within and outside the perturbative regime.
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### Review information

10.14293/S2199-1006.1.SOR-PHYS.A2CEM4.v1.RRHUMH

This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.

 ScienceOpen disciplines: Mathematical physics, Quantum physics & Field theory, Quantum gases & Cold atoms, Mathematical & Computational physics, Physics, Mathematics

### Review text

I found this paper an interesting work on perturbation theory. I think, for a better readership, the paper could benefit from
1. a brief discussion on the regime of linearity of perturbations and when this linearity breaks down and
2. a quantitative clarification of such propositions as assumptions being "only approximately satisfied", the level spacing and coupling strength being "slowly varying", etc...

As indicated by the authors' own admission, the results of the paper are relevant only to single particle models in one dimension. It might be worthwhile mentioning if there are any attempts/prospects to generalize this work.

I also wonder if using the normalized sinc function, rather than the sinc function, would be any more convenient.