Continuity of the integrated density of states

Slug: simon-ids-continuity
Status: residue — the general bounded continuum case in dimensions at least four remains open
Area: spectral theory, ergodic Schrödinger operators, quantitative unique continuation
Sources: Simon 2000, Bourgain–Klein, and Frank–Ivanisvili
Started: 2026-08-27

Result

No new continuity class is claimed. Bourgain and Klein already prove a deterministic log-Hölder estimate for every bounded continuum potential in dimensions one, two, and three. Their proof has a sharp dimensional frontier: its quantitative unique-continuation exponent is $4/3$ , and the argument requires

\frac{d}{d-1}>\frac{4}{3}.

The verifier in compute/q1/ replays the exact balance. For the published bounds in $d=2,3$ , the logarithmic exponent is

\kappa_d=\frac{4-d}{8}.

Its formal continuation is positive only for $d=2,3$ and is zero at $d=4$ . Frank and Ivanisvili's 2026 real-valued Landis examples rule out replacing $4/3$ by any smaller power in a uniform estimate of the same form. That closes this particular route through dimension four; it does not supply a discontinuous IDS.

One reusable special case was checked. If a lower-bounded ergodic operator $A$ is extended by $m\geq1$ free continuum directions,

H=A\otimes I+I\otimes(-\Delta_{\mathbb R^m}),

then its density-of-states measure is the convolution of the measure for $A$ with the free density of states. Consequently its IDS is locally absolutely continuous in every total dimension. This is a standard separable-operator consequence. It is recorded as a finite handle, and no dent follows from it.

The object Simon called $k(E)$

Let an ergodic group action translate a covariant family

H_\omega=-\Delta+V_\omega

on $L^2(\mathbb R^d)$ . Assume the family is lower bounded and has a locally finite density-of-states measure. For a unit cube $Q$ , the Pastur–Shubin trace formula is

\nu(B)=\mathbb E\!\left[ \operatorname{Tr}\bigl(\chi_Q\mathbf 1_B(H_\omega)\chi_Q\bigr) \right].

Simon writes the cumulative distribution as

k(E)=\nu(({-\infty},E]).

Equivalently, at continuity points it is the almost-sure thermodynamic limit of the number of finite-volume eigenvalues at most $E$ , divided by volume. On $\mathbb Z^d$ , the local trace is the expected spectral measure at one lattice site.

Continuity of $k$ is equivalent to the absence of atoms in the DOS measure. An atom at a fixed energy corresponds, in the ergodic setting, to an eigenspace with positive dimension per unit volume.

Published cases

Setting	Published conclusion
One-dimensional continuum	Craig–Simon log-Hölder continuity for bounded ergodic potentials.
Discrete, every dimension	Craig–Simon log-Hölder continuity; Delyon–Souillard give a short continuity proof.
Continuum, dimensions 2 and 3	Bourgain–Klein log-Hölder continuity for every bounded potential whenever the IDS exists.
Singular continuum potentials, dimensions at most 3	Klein–Tsang extend the density-of-states outer-measure estimate under their stated integrability hypotheses.
Anderson-type continuum models, every dimension	Wegner estimates give global continuity for atomless single-site laws and local continuity for arbitrary laws in localization regimes.
Periodic continuum models, every dimension	Floquet theory and the absence of flat bands give continuity.

The leftover addressed here is the continuum family with no independence, periodicity, localization, or separability assumption, already for bounded real potentials in dimension four.

A checked free-direction lemma

Let $A_\omega$ act on $L^2(\mathbb R^n)$ , be bounded below by $a$ , and have DOS measure $\nu$ .

Let $v_m$ denote the volume of the unit ball in $\mathbb R^m$ , and put

C_m=\frac{v_m}{(2\pi)^m},\qquad \alpha=\frac m2.

For the tensor-sum operator above, Fourier transformation in the free variables gives, first for compactly supported bounded Borel test functions and then for bounded energy intervals,

\nu_H(B)=\frac1{(2\pi)^m} \int_{\mathbb R^m}\int_{\mathbb R} \mathbf 1_B(\lambda+|p|^2)\,d\nu(\lambda)\,dp.

Tonelli's theorem and radial integration therefore give

k_H(E)=C_m\int_{(-\infty,E]}(E-\lambda)^\alpha\,d\nu(\lambda).

Thus the DOS is locally absolutely continuous, with an almost-everywhere density

n_H(E)=C_m\alpha\int_{(-\infty,E]}(E-\lambda)^{\alpha-1} \,d\nu(\lambda).

The apparent square-root singularity for $m=1$ is locally integrable. More explicitly, fix $E\leq E_0$ and $0<h\leq1$ . Put

M_0=\nu(({-\infty},E_0+1]).

For $m=1,2$ , concavity gives

0\leq k_H(E+h)-k_H(E)\leq C_mM_0h^{m/2} \quad(m=1,2).

For $m\geq3$ , the mean-value theorem gives

0\leq k_H(E+h)-k_H(E) \leq C_m\alpha M_0 R^{\alpha-1}h, \qquad R=\max\{1,E_0+1-a\}.

The certificate records the exponent $\min\{1,m/2\}$ . The convolution identity is standard for separable operators; the notebook therefore makes no priority claim for this lemma.

What would count as a new bound

A dent would be a checked continuity theorem that strictly enlarges a published class. Examples include continuity for arbitrary bounded ergodic continuum potentials in one new dimension $d\geq4$ , a new potential class not covered by the cited Wegner or separable results, or a strictly stronger certified modulus on a published class.

A complete proof must also identify the thermodynamic IDS with the local trace measure and handle every energy. An incomplete finite-volume search or an improvement of an auxiliary inequality that does not yield continuity is residue.

Acknowledgements

Stephen Wu selected the problem and set the verification standard. GPT-5.6 Sol performed the literature triage, obstruction analysis, free-direction derivation, and verifier implementation.