Exact Bases with Exponential-Polyhomogeneous Asymptotics

Definitions and mathematical statement

The supplied record concerns a family of ordinary differential operators \[ P_h=\sum_{k=0}^{m}C_k(x,h)\partial_x^k, \qquad h\to0^+. \] Here $m$ denotes the displayed differential order. The source refers to functions smooth on the interior of $I\times[0,h_0)$, where $I$ and $h_0$ are not further specified. The coefficient class, the meaning of "generalized semiclassical," and the hypotheses ensuring that the displayed operator has order $m$ are absent from the supplied excerpt.

An exact solution is a function satisfying $P_hu=0$. A quasimode is an approximate solution; the required size and topology of its residual $P_hq$ are not supplied here. The source introduces exponential-polyhomogeneous functions through a corrupted formula that appears to have the form \[ u(x,h)=e^{\phi(x)/h}a(x,h), \qquad a(x,h)\sim\sum_{j=0}^{\infty}h^{\gamma_j}a_j(x). \] This display is a tentative reading of the supplied transcription, not a recovered complete definition. In particular, the transcription contains an unexplained $\delta$, and does not reliably determine the exponential scale, admissible phases, exponent sequence, possible logarithmic terms, or remainder estimates. These details must be recovered from the cited report before this definition can be used precisely.

Conjectural target supplied by the record. For operators in the intended generalized semiclassical class, there exists a basis of exact solutions of $P_hu=0$ having the same exponential-polyhomogeneous asymptotic behavior as the quasimodes constructed in the source. Thus the requested correspondence is between exact solutions and the source's particular quasimodes, with their prescribed asymptotic data.

The supplied record does not identify those quasimodes or their construction. It also does not specify the solution space underlying $\ker P_h$, its scalar field, boundary conditions, whether the basis is local or global in $x$, or whether it is a basis for each fixed small $h$ or a basis in a space of parameter-dependent families. Consequently, neither the quantifiers over admissible operators nor the precise asymptotic matching condition can be completed from the supplied evidence. In particular, the statement must not be interpreted as a theorem for arbitrary coefficients $C_k$.

Short English statement

The conjecture asks whether the particular approximate solutions constructed in the source can be matched by a basis of genuine solutions with the same prescribed exponential and amplitude asymptotics as $h\to0^+$. The operator hypotheses, quasimode construction, and exact meaning of asymptotic agreement are missing or corrupted in the supplied excerpt, so a fully precise formulation requires human review of the cited report.

Sources

• Local UnsolvedMath registry record, permanent ID $30005602$, source problem code catalogue entry; registry locations identified by the project are data/unsolvedmath/problems.json and catalog/problems.json. This draft uses the supplied complete record without independently inspecting those files.
• Supplied report citation: Geometric Spectral Theory (2024), Oberwolfach Reports, pp. 2017--2106, DOI 10.4171/owr/2023/36. Supplied literature-source label: "Oberwolfach Report 36." https://doi.org/10.4171/owr/2023/36. No individual proposer is supplied. The record gives proposed year 2023; its report citation uses 2024. Both are retained without resolving their relationship.
• Supplied literature assessment, checked on 2026-08-22: the existence of an actual solution basis with the prescribed quasimode asymptotics is stated as a conjecture, and the checked source is reported to provide no proof. The supplied status is "open." This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims; this draft does not independently verify them or claim to have visited the source.