Attack log — continuity of the integrated density of states

Chronological attempts, newest last.

2026-08-27 — source and record

Opened Simon's 2000 reprint and checked the printed statement. His Problem 14 asks for continuity of $k(E)$ , records the one-dimensional and discrete cases, and singles out the higher-dimensional continuum case.
Reconstructed $k$ as the cumulative Pastur–Shubin local trace measure and separated the discrete, one-dimensional continuum, and multidimensional continuum settings.
The literature reset the starting line. Bourgain–Klein (2013) proves continuity for arbitrary bounded deterministic continuum potentials in $d=2,3$ , whenever the IDS exists. Specialized Wegner estimates cover many random models in every dimension. The unrestricted bounded continuum leftover begins at $d=4$ .

In Bourgain–Klein's proof, the harmonic-jet constraints leave a subspace when the vanishing order has scale
$N\asymp \rho^{1/(d-1)}R^{d/(d-1)},$
where $\rho$ is the finite-volume density in a short energy interval.
Their quantitative unique-continuation estimate costs $R^{4/3}$ . The needed comparison $N>MR^{4/3}$ is available only when
$\frac d{d-1}-\frac43>0.$
Solving that comparison for $\rho$ reproduces $\rho\lesssim R^{(d-4)/3}$ . Their remaining choice of scale gives the logarithmic modulus exponent $(4-d)/8$ .
compute/q1/verify_frontier.py and an independent Rust verifier check the reduced fractions from certificates. Both identify equality and a zero exponent at $d=4$ .

Bourgain–Klein explicitly observed that a power $\beta<4/3$ in place of $4/3$ would extend their method to dimensions satisfying
$d<\frac{\beta}{\beta-1}.$
Frank–Ivanisvili (arXiv:2608.00802) now construct, for every $d\geq3$ , smooth real $u\ne0$ and smooth bounded real $V$ with
$-\Delta u+Vu=0,\qquad |u(x)|\leq C\exp(-c|x|^{4/3}).$
This construction rules out the required power improvement for a uniform estimate of Bourgain–Klein's form. Fix a ball where $u$ has positive mass and test a unit ball at distance $Q$ . A hypothetical $Q^\beta$ estimate with $\beta<4/3$ has a lower side of order
$\exp(-C Q^\beta\log Q),$
while the constructed solution makes the observation-ball mass at most $\exp(-cQ^{4/3})$ . The latter is eventually smaller, a contradiction.
The example has no ergodic structure and supplies no IDS counterexample. It is a sharp obstruction to lowering the power in this deterministic unique-continuation input. Other proof architectures remain open.

Delyon–Souillard bound a discrete eigenspace restricted to a cube by the number of values in two boundary layers, $O(L^{d-1})$ . Dividing by the volume rules out a DOS atom.
Continuum Cauchy data on a hypersurface are infinite-dimensional, so the same count has no finite boundary handle. Replacing boundary values by harmonic jets returns exactly the $N^{d-1}$ codimension count used by Bourgain–Klein, together with its quantitative propagation cost. This route therefore meets the same dimension frontier.

For
$H=A\otimes I+I\otimes(-\Delta_{\mathbb R^m}),$
Fourier transform in the free variables gives the exact DOS identity
$\nu_H=\nu_A*\nu_{\mathrm{free},m}.$
The free DOS has the locally integrable density proportional to $t_+^{m/2-1}$ . Hence the tensor-sum IDS is locally absolutely continuous even when the base measure has atoms. The formulas and local Hölder/Lipschitz exponents are proved in PROBLEM.md and replayed in compute/q1/.
The convolution identity is standard separable-operator theory and appears explicitly in later DOS literature. It is a useful exact class. The published record therefore receives no dent.

Residue: no continuity theorem beyond a published class was found.
Verified handle: the Bourgain–Klein method works precisely for $d<4$ , is critical at $d=4$ , and cannot be extended merely by replacing the real-potential Landis exponent with a power below $4/3$ .
Leftover: continuity for general bounded ergodic continuum Schrödinger operators in every dimension $d\geq4$ .