# Talk:Spectral properties of quantum diffusion

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## Reviewer A: review

This article is a very good summary of what is known right now about quantum diffusion in homogeneous media and its relation with the fractal properties of the energy spectrum. In real materials, such as crystals in a uniform magnetic field, semiconductors at low temperature,or quasicrystals, this type of diffusion is dampen by dissipation effect, but it can be seen through the electric conductivity as the temperature goes to zero. This is because the dissipative effect become less and less important as the temperature decreases, unfolding the complexity of quantum effects.

As very well explained in this article, the heuristic argument leading to a relationship between fractal properties of the energy spectrum and the time behavior of a typical wave packet is based upon the Heisenberg uncertainty relation. This latter expresses that it is not possible to discriminate energy at a scale 1/t by observing the wave packet evolution only on a time less than t. Then, the article comes to give the most significant rigorous inequalities, while mentioning the existence of few finer results that were obtained overtime. The list of reference is reasonable for this type of article for a general public of scholars.

What could have been added to this article, however, is the consequence of this analysis on the low temperature behavior of the electric conductivity, leading to an anomalous temperature dependence according to an "anomalous Drude formula". The latter is actually important in filling the gap between the abstract mathematical analysis, presented in the present article, and the experimental observations. The first example of experiment exhibiting such a behavior were conducted by Claire Berger in Grenoble between 1987 and 1990 on quasicrystals and showed very precisely such an anomalous behavior, unknown at the time, and that can be explained well through this anomalous Drude formula. But that topics can be the subject of a separate article.

Italo Guarneri, the author of this article, gave the first rigorous treatment of this effect in 1989 and can be considered as the leading expert in this matter. He contributed to the improvement of the bounds at several occasions during the twenty years that followed his first paper. He is therefore the best choice for an article of that nature.

## Reviewer B: review

It is a well written physics article by the top expert in the field, but at the moment it misses some of the important rigorous developments, mainly of the last decade. It also is a bit vague in places, from the rigorous standpoint, but that's probably OK.

Overall, I think it is important to stress more strongly (perhaps through several examples) that it is not just/necessarily the fractal nature of the spectrum as a set (or even the support of the spectral measure) that is directly relevant to anomalous quantum diffusion but an interplay between the properties of spectra and spectral projectors.

The results that, I think, need to be mentioned (expanding the exact results section) and that would enhance readers' understanding, include (the list is, more or less, chronological):

a) the obvious ballistic upper bound

b) Avron-Simon's transient and recurrent spectrum

c) Simon's absence of ballistic motion for point spectrum

d) del Rio-Jitomirskaya-Last-Simon (1996) example of point spectrum with localized eigenfunctions and motion, as strong as almost ballistic

e) Random dimer model, where this effect (spectral localization, yet superdiffusive anomalous transport with an exact exponent) occurs for a physically relevant model (Dunlap-Wu-Phillips, 1990, and Germinet-DeBievre (rigorous localization) Jitomirskaya-Schulz-Baldes-Stolz and Jitomirskaya-Schulz-Baldes ( rigorous transport))

f) Upper bounds in 1D of Damanik-Tcheremchantzev and their application to Fibonacci

Killip-Kiselev-Last also could be mentioned along with Kiselev-Last.

Finally, the link to Hofstadter butterfly is outdated (e.g. the linked site lists some solved problems as open, and I doubt it's been maintained for the last 5 years).

## Reply to Reviewer A

ok. Thanks for the suggestion.

## Reply to Reviewer B

Thanks: I've accepted most suggestions. The following remarks are meant to simplify the next round. The Editors' Instructions invite not to presume too much of a specialistic culture of the readers, and to minimize references to specialistic papers. This article in the Quantum Chaos category is theoretical-physics oriented and I do not presume a generic theor-phys graduate student to know enough about rigorous spectral theory to understand much of several rigorous results. I don't think this Scholarpedia article to be a proper place, nor myself a qualified author, to supply such spectral knowledge. So I've resorted to a number of shortcuts/compromises/bypasses. I consider this an acceptable price to pay for communicating the general ideas, which however does not afford venturing farther in the way of mathematical precision/completeness.

On such grounds, I had not mentioned several relevant specialistic works, including some of my own. I may have exaggerated and I've now included most suggestions by B. Not all of them, though:

- del Rio et al.'s example is very technical and somehow physically artificial, I think the dimer model alone does the job here. (Omitting it was a real mistake and I'm indebted to B for pointing it out).

- Having bypassed a proper definition of an ac spectrum, it seems hardly appropriate to mention Avron and Simon's technical distinction between transient and recurrent ac spectra.

- For similar reasons it seems hardly appropriate to go into more details about the contents of old/new references.

- Hofstadter: that link is not meant to update the reader to the state of the specialistic art concerning Harper, but just to provide a smooth introduction to it, and in that sense I find it very good. I'd suggest keeping it, possibly with an explicit caveat. However I'm not a specialist in that model and we should be indebted to B for indicating any reference/link that may conjugate timeliness and Scholarpedia style.

## Reviewer B:

The new edition does a good job in addressing the issues I brought up in the review. there is one new technical comment: the discussion about the ballistic bound in the last paragraph, as currently presented, is only correct for $d=1.$ I suggest this should be modified to include all dimensions. Namely, the general ballistic bound is by $t^{2r},$ so by (3) it is only necessarily attained with ac spectra for $d=1.$ It may also be good to mention here that it is expected that ac spectrum in the Anderson model with $d\ge 3$ is diffusive.

## Answer to Reviewer B:

That's a nice suggestion indeed, however I'd rather not go into that, for the time being. Otherwise I'd have to explain what "expected" means. I refrain from this for already explained reasons.