Spectrum of Boundary Problems with Noncompact Multiplier Weights

Definitions and mathematical statement

Let $n$ denote the number of terms in the self-similar representation of a function $P$. Assume, in the terminology of the supplied source, that $P$ has zero spectral order and that its generalized derivative $P'$ is a noncompact Sobolev multiplier. Consider the singular Sturm--Liouville boundary-value problem \[ -y''=\lambda P'y,\qquad y(0)=y(1)=0. \] Here $\lambda$ is the spectral parameter. The precise function spaces, meaning of the multiplier action $P'y$, and operator or operator-pencil realization are not specified in the supplied record.

The requested problem is to determine the complete spectrum of this boundary-value problem, including its spectral type and eigenvalue behavior. More specifically, the supplied formulation asks whether, for every $P$ satisfying the stated hypotheses, the spectrum consists of $n-1$ possibly overlapping intervals of continuous spectrum together with eigenvalues, with at most one eigenvalue in each gap and at most $n-2$ eigenvalues in total.

This is a conjectural description, not an asserted spectral theorem. The supplied record does not define an $n$-term self-similar function, zero spectral order, or the Sobolev spaces involved in the noncompact multiplier hypothesis. It also leaves unspecified the admissible self-similarity parameters, any reality or sign assumptions, the spectral realization, interval endpoint conventions, the meaning of a gap, and whether eigenvalues are counted with multiplicity. Consequently, the displayed problem and proposed spectral description retain the source terminology and require these qualifications to be resolved before becoming a fully specified mathematical assertion. The allowed range of $n$ also requires clarification, since the proposed upper bound $n-2$ is negative when $n=1$.

Short English statement

For a self-similar weight whose generalized derivative is a noncompact Sobolev multiplier, determine the spectrum of the stated Dirichlet problem. The supplied conjecture predicts $n-1$ possibly overlapping continuous-spectrum intervals, at most one eigenvalue in each gap, and at most $n-2$ eigenvalues altogether. Essential definitions and realization details remain unresolved; this is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004276$, title "Spectrum of Boundary Problems with Noncompact Multiplier Weights," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied primary-source citation: Mini-Workshop: Self-adjoint Extensions in New Settings (2020), Oberwolfach Reports, pp. 2911--2950, DOI 10.4171/owr/2019/47. https://doi.org/10.4171/owr/2019/47. The supplied citation uses 2020, whereas the DOI and proposed-year metadata contain 2019; their relationship has not been independently checked.
• Supplied literature source: "Sharov--Sheipak, string equation with a noncompact multiplier." https://www.mathnet.ru/php/archive.phtml?issue=4&jrnid=aa&option_lang=eng&volume=33&wshow=issue&year=2021. The supplied link identifies a journal issue rather than a complete article citation.
• Supplied literature source: "original Oberwolfach report." https://doi.org/10.4171/owr/2019/47.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that later work determines continuous spectrum and eigenvalue behavior for important noncompact-multiplier string models, while the general complete spectral description remains unestablished. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified and reports that the official report was checked to replace a dangling boundary-value reference with the displayed equation, boundary conditions, self-similar weight hypothesis, and spectral conjecture. These are supplied verification claims, not verification performed for this draft.