Phase-Space Quantization for Variable-Coefficient PDEs

Definitions and mathematical statement

The supplied fragment concerns a differential operator $P(D)$ with constant coefficients and polynomial symbol, transcribed from the damaged summation as \[ p(\xi)=\sum_{|\alpha|\leq N}c_\alpha\xi^\alpha. \] In this notation, $\alpha=(\alpha_1,\ldots,\alpha_n)$ is a multi-index of nonnegative integers, $|\alpha|=\alpha_1+\cdots+\alpha_n$, and $\xi^\alpha=\prod_{j=1}^n\xi_j^{\alpha_j}$. The dimension $n$, order bound $N$, coefficient field, underlying domain, and convention for $D$ are not specified in the fragment. The displayed polynomial is a transcription of the constant-coefficient example, not a supplied formulation of the variable-coefficient problem. The equations mentioned are \[ P(D)u=0 \qquad\text{or}\qquad P(D)u=\text{``small''}. \] No norm, topology, scale, or parameter defining "small" is supplied.

Recoverable question. For partial differential equations with nonconstant coefficients, can the phase-space maps referred to in the surrounding discussion be quantized so that they act at the level of operators?

Here "quantized" retains the source's informal meaning of passing from phase-space maps to operators; the fragment supplies no mathematical definition or required operator identity. It does not identify the maps, their domains, the class of differential operators, or the function spaces on which a quantization should act. In particular, "any of the phase space maps" does not determine a universal quantifier over all phase-space maps. Whether the question concerns local or global constructions, exact identities or asymptotic relations, or any prescribed boundary conditions is unresolved. Consequently, the supplied evidence does not determine a precise existence theorem or conjecture.

The supplied status is partially_solved, and the statement is marked unrecoverable. The supplied dated literature assessment says that semiclassical Fourier integral operators give local quantizations when the characteristic set is a graph, while the fragment does not establish a construction for arbitrary variable-coefficient equations or global gluing. A separate supplied verification note says that the official report immediately answers the question using a semiclassical Fourier-integral-operator construction and that it is therefore not an open problem. These statements are supplied curation, not independently verified mathematical conclusions; their scopes cannot be reconciled from the fragment alone.

Short English statement

The fragment asks whether phase-space maps for variable-coefficient partial differential equations can be represented by operators. It omits the maps, hypotheses, and meaning of quantization needed for a precise mathematical statement. The supplied partial-solution assessment and the separate claim that the report already answers the question require reconciliation.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30005239$, title Phase-Space Quantization for Variable-Coefficient PDEs, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year $2022$ and names no proposer.
• Supplied report citation: At the Interface between Semiclassical Analysis and Numerical Analysis of Wave Scattering Problems (2023), Oberwolfach Reports, pp. 2511--2587, DOI 10.4171/owr/2022/43. Supplied source URL: https://doi.org/10.4171/owr/2022/43. The background identifies Oberwolfach Reports as records of mathematical workshops and their open-problem discussions; no passage-level location for this fragment is supplied.
• Supplied literature source: Oberwolfach Report 43/2022, https://doi.org/10.4171/owr/2022/43. Literature checked at: 2026-08-22. This date and the associated partial-solution assessment are supplied dated curation, not independently verified. The background explicitly calls the assessment dated literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field claims that the report immediately answers the question through a semiclassical Fourier-integral-operator construction. This attribution is preserved as a supplied claim; the report has not been independently consulted for this draft.