Research log — continuity of the integrated density of states

Status cutoff: 2026-08-27. The arXiv versions are treated as the record when they exist. Older foundational papers predate arXiv; the journal or author archive is linked.

2026-08-27 — statement and definitions

• Barry Simon, Schrödinger operators in the twenty-first century, Mathematical Physics 2000, pp. 283–288. The Caltech reprint was downloaded and page 5 was rendered. Problem 14 says to prove that $k(E)$ is continuous, then records the one-dimensional and discrete cases and the open higher-dimensional continuum case.
• The ResearchGate landing page was also opened while locating the reprint. Its PDF link returned 404, so no claim is taken from it.
• Werner Kirsch and Bernd Metzger, arXiv:math-ph/0608066, archived full PDF, The Integrated Density of States for Random Schrödinger Operators. This survey gives the local trace definition, the thermodynamic-limit framework, the Delyon–Souillard boundary argument, and the regularity landscape as of 2006.

2026-08-27 — foundational record

• Leonid Pastur, Spectral properties of disordered systems in the one-body approximation, Comm. Math. Phys. 75 (1980), 179–196. This is a foundational ergodic random-operator source. The Springer page was readable; its full PDF is subscription-gated.
• M. A. Shubin, The spectral theory and the index of elliptic operators with almost periodic coefficients, Russian Math. Surveys 34 (1979), 109–157. The Math-Net record links the English PDF and Shubin's earlier density-of-states work. The PDF endpoint timed out in the browser and returned 403 to the command-line fetcher.
• Walter Craig and Barry Simon, Subharmonicity of the Lyaponov index, Duke Math. J. 50 (1983), 551–560. The paper treats bounded one-dimensional ergodic continuum operators and their discrete analogues; the Thouless/subharmonicity argument gives log-Hölder regularity. The author-uploaded landing page was used to inspect the opening definitions when Project Euclid's iframe blocked text extraction.
• Walter Craig and Barry Simon, Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices, Comm. Math. Phys. 90 (1983), 207–218. This is the all-dimensional discrete log-Hölder record cited by Bourgain–Klein. Project Euclid's PDF endpoint returned an anti-bot page.
• François Delyon and Bernard Souillard, Remark on the continuity of the density of states of ergodic finite difference operators, Comm. Math. Phys. 94 (1984), 289–291. The discrete proof bounds an eigenspace in a cube by two boundary layers. Project Euclid's PDF endpoint returned the same anti-bot page; the theorem and proof were independently read in the Kirsch–Metzger survey.

2026-08-27 — the general deterministic advance

• Jean Bourgain and Abel Klein, arXiv:1112.1716v3 abstract and full HTML, Bounds on the density of states for Schrödinger operators, Invent. Math. 194 (2013), 41–72. Theorem 1.1 proves, for every bounded deterministic continuum potential in $d=1,2,3$,

  $$\eta^*([E,E+\varepsilon]) \leq C\bigl(\log(1/\varepsilon)\bigr)^{-\kappa_d}, \qquad \kappa_1=1,\quad\kappa_2=\tfrac14,\quad\kappa_3=\tfrac18.$$

  The outer measure dominates any DOS measure that exists, so this proves IDS continuity for ergodic families in those dimensions. Equations (3.95), (3.111), and (3.115)–(3.121) expose the dimension balance. The introduction says that a UCP power $\beta$ would allow dimensions below $\beta/(\beta-1)$; their $\beta=4/3$ gives $d<4$.

• Abel Klein and C. S. Sidney Tsang, arXiv:1408.7111, Local behavior of solutions ... with singular potentials and bounds on the density of states. This extends the outer-measure result to stated singular potential classes in dimensions one through three. It does not cross the dimension-four frontier.

2026-08-27 — random classes in every dimension

• Jean-Michel Combes, Peter Hislop, and Frédéric Klopp, arXiv:math-ph/0605029v2, An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators, Duke Math. J. 140 (2007), 469–498. For continuum Anderson-type models in every dimension, a compactly supported nonnegative single-site potential and an atomless conditional law give global continuity. A bounded density gives a locally Lipschitz IDS.
• The earlier Combes–Hislop–Klopp article, Hölder continuity of the integrated density of states for some random operators at all energies, IMRN 2003, 179–209, gives all-energy Hölder regularity under a bounded-density Anderson hypothesis.
• François Germinet and Abel Klein, arXiv:1105.0213v3, A comprehensive proof of localization for continuous Anderson models with singular random potentials, JEMS 15 (2013), 53–143. It allows arbitrary bounded nondegenerate single-site distributions, including Bernoulli, and proves log-Hölder IDS continuity at the bottom of the spectrum in the localization region.
• Thomas Hupfer, Hajo Leschke, Peter Müller, and Simone Warzel, arXiv:math-ph/0105046v2, The absolute continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded random potentials, Comm. Math. Phys. 221 (2001), 229–254. A one-parameter decomposition gives local Lipschitz continuity for broad alloy and Gaussian examples, including some unbounded potentials.
• Norbert Peyerimhoff and Ivan Veselić, arXiv:math-ph/0210047, concerns existence of the IDS on amenable covering manifolds. It was checked for the definition/existence side and does not supply general continuity.
• S. Doi, A. Iwatsuka, and T. Mine, arXiv:math-ph/0010013, concerns existence and uniqueness with magnetic fields and unbounded random potentials. It is an existence source and leaves Simon's continuity question open.

2026-08-27 — obstruction and separable handle

• Rupert Frank and Paata Ivanisvili, arXiv:2608.00802v1 abstract and full HTML, Counterexamples to the Landis conjecture in dimensions three and higher. Theorem 1 constructs smooth real bounded $V$ and a nonzero smooth real solution decaying like $\exp(-c|x|^{4/3})$ in every $d\geq3$. This makes the power needed by the Bourgain–Klein extension impossible in a uniform estimate of the same form. The paper does not discuss an IDS or an ergodic family.
• The two-dimensional contrast is Logunov, Malinnikova, Nadirashvili, and Nazarov, The Landis conjecture on exponential decay, Invent. Math. 241 (2025), 465–508. It supports a weaker positive Landis statement in the plane and does not alter the $d\geq4$ IDS question.
• David Damanik and Anton Gorodetski, Spectral transitions for the square Fibonacci Hamiltonian, J. Spectr. Theory 8 (2018), 1487–1507. Equations (1)–(2) explicitly call the spectrum-sum and DOS-convolution identities standard separable-operator theory. This is the precedence check that prevents treating the free-direction convolution calculation as a new continuity class.

Fetch notes

• python3 scripts/arxiv_fetch.py 1112.1716 --research problems/simon-ids-continuity/RESEARCH.md first failed because the sandbox could not resolve the host. The approved retry reached the arXiv API and returned HTTP 429. The abstract and complete v3 HTML were therefore read directly at the links above.
• The same fetcher was tried for math-ph/0608066; the approved retry also returned HTTP 429. Its arXiv abstract and the archived full survey PDF linked above were read instead.
• Direct command-line fetches of the three Project Euclid originals returned anti-bot HTML. The Math-Net English-PDF endpoint returned HTTP 403. These failures are recorded so that the accessible journal pages are not mistaken for locally replayed PDFs.