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This question was posted a week ago on MathOverflow https://mathoverflow.net/q/369156/

The Anderson Model is given by the random Hamiltonian (as an operator on $l^2(\mathbb{Z}^d)$)

$$ H_\omega = - \Delta + V(\omega) $$ where $V(\omega) \mid x \rangle = \omega(x) \mid x \rangle$ with $\{ \omega(x) \}_{x \in \mathbb{Z}^d}$ independent and uniformly distributed in $[-L, L]$.

It is known for example by Kunz and Soulliard that the spectrum $ \sigma(H_\omega) = [-L, 4d +L ]$. This is also easily verified by simulating finite dimensional approximations. By simulation it is also easy to look at the density of states. Below I have plotted the density of states for $L=0,1,5$ for a $500 \times 500$ approximation. One can see that the high probabilities towards the end of the spectrum wash out. So far this is all numerics, but what is known analytically about this particular density of states?

The density of states plotted for <span class=$L=0,1,5$ for 500x500 approximatio" />

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    $\begingroup$ We've written a paper on it, it's currently under peer review. I can answer this in a few months if nobody here shows that our paper is trivial :) $\endgroup$
    – Razor
    Commented Aug 22, 2020 at 1:46
  • $\begingroup$ Interesting! Aren't there any conjectures for exact formulas in the literature? What can you say about the average distance to the origin? Feel very free to send me the paper once it is possible! $\endgroup$ Commented Aug 22, 2020 at 16:11
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    $\begingroup$ @Razor: Is your paper out now? $\endgroup$ Commented Nov 24, 2020 at 20:07

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A great (and mathematically rigorous) review on Anderson's model can be found in Aizenman's textbook, Random Operators: Disorder Effects on Quantum Spectra and Dynamics. See CHapter 3 and 4 for discussions/bounds on density of states.

And just a side note, it's actually pretty difficult to numerically verify that $\sigma(H_\omega) = [-L,4d+L]$ as explained by Corollary 4.16 in Aizenman's textbook.

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  • $\begingroup$ Thanks! That reference was actually my starting point. I am more interested averaging over the entire spectrum for say some Gibbs states and getting some way to do that. However, I wonder why physicists do not have some exact conjectures on the density of states? And yes, good point with the numerical verifications - you need to take care about the Lifshits tails! $\endgroup$ Commented Aug 22, 2020 at 16:08
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    $\begingroup$ I'm not quite sure what you mean by averaging over the spectrum, but if you want to formulate the DOS using Gibbs states $e^{-\beta H}$ instead of $\mathbb{E}\langle 0|f(H)|0\rangle$, maybe this reference may help. Pastur, L. A. Behavior of some Wiener integrals as t→∞ and the density of states of Schrödinger equations with random potential $\endgroup$ Commented Aug 22, 2020 at 20:28

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