Rigorous Boundary-Layer Density Asymptotics

Definitions and mathematical statement

The supplied record concerns slow densities denoted by $\eta_m$ along periodic multiple-scattering orbits, with the Dirichlet boundary condition. The excerpt also uses the notation $\eta_m^{\mathrm{slow}}(x,k)$. It does not define the scattering problem, the orbit index $m$, the variable $x$, the parameter $k$, or the relation between these density notations.

Requested statement, subject to unresolved source specifications. Establish rigorously that the asymptotic rate-of-convergence formulas derived for the slow densities along periodic multiple-scattering orbits describe the actual densities of the underlying Dirichlet scattering problem.

The damaged source excerpt indicates an expansion with terms $a_{p,q,r,\ell,m}(x,k)$ indexed by \[ p,q,r\geq 0,\qquad \ell\leq -1, \] and expressions involving $b_{p,q,r,\ell,m}(x)$, $(\Psi_{r,\ell})^{(p)}$, and $Z_m(x)$. The index domains beyond these inequalities, the meanings of these objects, and the powers and grouping involving $k$ are not reliably specified. Consequently, no exact expansion is asserted here.

A precise theorem requires the complete formulas and their hypotheses, the limiting regime, admissible geometries and orbits, domains of the variables, the meaning of asymptotic approximation, truncation rules, remainder estimates, and any uniformity in $x$ or $m$. These data are absent from the supplied record. In particular, it is unclear whether the requested convergence concerns the asymptotic parameter, successive scattering iterates, or both. This draft preserves the requested connection between formal formulas and actual densities without supplying missing mathematical assumptions.

Short English statement

Prove that the derived asymptotic formulas for slow densities along periodic multiple-scattering orbits describe the actual densities for the Dirichlet problem. The supplied excerpt is damaged and lacks the definitions, hypotheses, and error estimates needed to state a precise theorem. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005240$, title Rigorous Boundary-Layer Density Asymptotics; source problem code catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2022$; no individual proposer is identified.
• Supplied report citation: At the Interface between Semiclassical Analysis and Numerical Analysis of Wave Scattering Problems (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/43, pp. 2511--2587. Supplied source URL and literature-source link, labeled Oberwolfach Report 43/2022: https://doi.org/10.4171/owr/2022/43. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that a completely rigorous proof relating the asymptotic formulas to the actual densities $\eta_m$ remains open. This is supplied dated curation, not independently verified; the record itself requests independent verification of time-sensitive status and cited claims.
• The supplied record labels its statement corrected_verified and says it was reconstructed and checked against the official report. That verification claim is supplied provenance, not a source consultation or mathematical verification performed for this draft.