Higher-Order Correlations at Coincident Energies

Definitions and mathematical statement

Let $\mu_r$ denote the source's order-$r$ spectral correlation measure, with $r\geq2$, for its random Schrödinger model. Such measures depend on the self-adjoint random operator, inserted observables and trace/volume normalization; the excerpt supplies none of those definitions. They must not be replaced by independent products of the density of states or assumed to be positive probability measures solely from the word correlation.

Write \[ \Sigma_r=\bigcup_{1\leq j<k\leq r}\{E\in{\mathbb{R}}^r:E_j=E_k\}, \qquad U_r={\mathbb{R}}^r\setminus\Sigma_r. \] The assessment reports an analytic density $\rho_r$ away from coincidence planes in the appropriate strong-disorder setting. In that specified regime one has the off-diagonal representation $d\mu_r=\rho_r(E)\,dE$ on the applicable part of $U_r$. The question asks for behavior near and on $\Sigma_r$, including the planes and their multiple intersections, rather than another off-diagonal analyticity statement.

This requires a precise interpretation. A Radon--Nikodym density relative to $r$-dimensional Lebesgue measure is only defined almost everywhere, and its values on the null set $\Sigma_r$ are arbitrary without an extension/trace convention. Meaningful targets could therefore concern limits as energy gaps vanish, boundedness or singular asymptotics, extendibility/differentiability across the planes, or a measure component supported on them. These are distinct questions. Off-diagonal analyticity alone neither fixes diagonal values nor excludes diagonal-supported singular measures.

The source does not specify a desired regularity order, leading rate, one-sided approach, treatment of intersecting coincidence planes, or a global absolute-continuity assertion. It also omits the operator dimension, disorder law/strength and observables of the known theorem. No numerical singularity exponent or absence of singular part is invented.

The literature triage checked 21 August 2026 says the 2007 result proves analyticity away from coincident planes and explicitly leaves their near/on behavior open; it reports no general higher-order resolution in the later cited work. Those theorem-scope and status statements are retained as dated supplied information, not independently verified conclusions for every random Schrödinger model.

Short English statement

Describe higher-order spectral correlations as energies coincide, beyond the known off-diagonal analytic regime. The issue may involve limits, regularity or singular measure components; a density's arbitrary pointwise values on a null hyperplane are not themselves defined data. The exact model and target behavior are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000710, exact source code catalogue entry; Higher-Order Correlations at Coincident Energies. Proposed year 2007, status open; corrected_verified editing preserves the density/near-and-on-coincidence-plane question.
• Transport in Multi-Dimensional Random Schrödinger Operators (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/12, pp. 667--708, https://doi.org/10.4171/owr/2007/12.
• Supplied literature triage checked 2026-08-21: "source report", https://doi.org/10.4171/owr/2007/12; "strong-disorder regularity theorem", https://doi.org/10.1007/s00023-006-0297-1. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; the precise model/theorem hypotheses are not reproduced.