Unlike Part I, I will now do lots of math, and I will derive the
central formula in:
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.050401
which was found by Raphael Ribeiro.
Any appearance of rigor will be an illusion.

More than 30,000 papers are published each year
in which modern density functional calculations are performed.
However, there is presently no systematic route to finding useful
approximations. Over 40 years ago, Lieb and Simon demonstrated
that the original version of density functional theory, Thomas-Fermi
theory, becomes relatively exact in a very particular
non-relativistic limit of large electron number. I will explain why I believe this
holds the key to a systematic treatment of such approximations, and
what my group has done in the last 8 years to use this insight.

Abstract:
 We consider decaying oscillatory perturbations of periodic Schr\"odinger
 operators on the half line. More precisely, the perturbations we study
 satisfy a generalized bounded variation condition at infinity and an $L^p$
 decay condition. We show that the absolutely continuous spectrum is
 preserved, and give bounds on the Hausdorff dimension of the singular part
 of the resulting perturbed measure. Under additional assumptions, we
 instead show that the singular part embedded in the essential spectrum is
 contained in an explicit countable set. Finally, we demonstrate that this
 explicit countable set is optimal. That is, for every point in this set
 there is an open and dense class of periodic Schr\"odinger operators for
 which an appropriate perturbation will result in the spectrum having an
 embedded eigenvalue at that point.

A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520 . . . (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refinements of the AM-GM inequality, known as Maclaurin’s inequalities, relating the 1/kthpowers of the kth elementary symmetric means of n numbers for 1 ≤ k ≤ n. On the left end (when k = n) we have the geometric mean, and on the right end (k = 1) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves f (n) steps away from either extreme. We also study the limiting behavior of such means for quadratic irrational α.

(Joint work with Francesco Cellarosi, Doug Hensley and Steven J. Miller)