One of interesting properties of Schroedinger operators $H$ with
periodic and almost periodic potentials is that for an auxiliary
Hamiltonian restricted to an infinitely deep d-dimensional box and
for the states which lie below an energy-cutoff $\lambda$ there
exists an infinite-volume limit $D(\lambda)$ of their number per
volume (called the density of states).

The problem addressed in the paper is an asymptotic estimate of
the smallness of the difference $D(\lambda)-\lambda/(4\pi)$
(and/or of its trivial modifications) at $d=2$. The result (viz.,
its proportionality to $\lambda^{-6/5+ something}$ formulated as
Theorem 2.3) is impressive.

The technical text itself presents the most important
partial-differential generalization of the recent ordinary
differential $d=1$ result by the same author (ref. [25], to appear
in 2005). In this setting, while the basic idea of the $d=1$ proof
lied in a ``gauge" transformation of $H$ into an operator with
constant coefficients, the key novelty of the present construction
lies in an adaptation of such a trick to $d\geq 2$. Naturally
(and, in some preparatory lemmas, manifestly), the possibility of
a future extension of the present result to $d>2$ is kept in mind.


MR2119355
Sobolev, Alexander V. Integrated density of states for the periodic
Schrödinger operator in dimension two. Ann. Henri Poincaré 6 (2005),
no. 1, 31--84.