Self-Adjointness for Degenerate No-Tangency Metrics

Definitions and mathematical statement

The supplied record does not specify the domain of the parameter $\ell$. The following gives a precise formulation for the positive-integer interpretation; whether this is the full intended parameter range requires source review.

For each $\ell\in{\mathbb{Z}}_{\geq 1}$, set \[ a_\ell(x,z)=x(x^{2\ell}+z^2),\qquad X_1=\partial_x,\qquad X_2=a_\ell(x,z)\partial_z. \] The regular region, where these vector fields are linearly independent, is \[ M_\ell=\{(x,z)\in{\mathbb{R}}^2:a_\ell(x,z)\ne0\}=\{(x,z)\in{\mathbb{R}}^2:x\ne0\}. \] Interpret the associated Riemannian metric as the metric making $X_1,X_2$ an orthonormal frame on $M_\ell$. Thus \[ g_\ell={\,\mathrm{d}} x^2+\frac{{\,\mathrm{d}} z^2}{a_\ell(x,z)^2},\qquad {\,\mathrm{d}}\mu_\ell=\frac{{\,\mathrm{d}} x\,{\,\mathrm{d}} z}{|a_\ell(x,z)|}. \] Use the Laplace--Beltrami convention \[ \Delta_{g_\ell}f =|a_\ell|\,\partial_x\!\left(|a_\ell|^{-1}\partial_x f\right) +|a_\ell|\,\partial_z\!\left(|a_\ell|\partial_z f\right). \] Let $H_\ell=-\Delta_{g_\ell}$ be the densely defined symmetric operator in the complex Hilbert space $L^2(M_\ell,\mu_\ell)$, with initial domain \[ \mathcal D(H_\ell)=C_c^\infty(M_\ell;{\mathbb{C}}). \] Here the entire regular region, including both components $x>0$ and $x<0$, is used. No boundary condition at the omitted line $x=0$, or at infinity, is imposed on this initial operator. Essential self-adjointness means that its closure is self-adjoint, equivalently $\overline{H_\ell}=H_\ell^*$.

Question, under the positive-integer interpretation. Is $H_\ell$ essentially self-adjoint for every $\ell\in{\mathbb{Z}}_{\geq1}$?

The metric, volume measure, and initial operator domain above make explicit the conventional interpretation of the supplied Laplace--Beltrami question; the record itself does not state these choices explicitly. Its notation $\ell\geq1$ does not specify whether noninteger parameters are intended. Such an extension requires a definition of $x^{2\ell}$ for negative $x$ and appropriate regularity assumptions, which are not supplied. No convention such as replacing $x^{2\ell}$ by $|x|^{2\ell}$ is adopted here.

The supplied dated assessment labels the problem partially solved and reports that the case $\ell=1$ is proved, whereas the corresponding assertion for all positive $\ell$ was expected but not proved in the report. These are supplied literature claims, not independently verified conclusions. The original record also mentioned a preceding real-analytic no-tangency conjecture; the supplied statement-verification note says that this broader question duplicated $\texttt{OWR-17290-001}$ and was removed. It is therefore not included as a second target here.

Short English statement

For each positive integer $\ell$, give the region $x\ne0$ the Riemannian metric for which $\partial_x$ and $x(x^{2\ell}+z^2)\partial_z$ are orthonormal. Ask whether its Laplace--Beltrami operator on compactly supported smooth functions is essentially self-adjoint. The intended parameter domain needs verification. Supplied dated curation reports a proof for $\ell=1$ and labels the general question partially solved; this assessment has not been independently verified.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004272$, title Self-Adjointness for Degenerate No-Tangency Metrics, source problem code catalogue entry, in Oberwolfach Reports Open Problems. The record supplies proposed year 2019 and no named proposer. Its statement-status label is corrected_verified; that label is supplied registry metadata, not independent verification of this draft.
• Supplied report 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 record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "OWR 2019 47": https://ems.press/content/serial-article-files/46828.
• Supplied literature source, attributed to Franceschi--Prandi--Rizzi: https://doi.org/10.1007/s11118-018-09760-w. No article title, theorem number, or precise supporting passage is supplied.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It reports that the explicit family is proved for $\ell=1$, while the report expected but did not prove the analogous essential-self-adjointness result for all positive $\ell$. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the general no-tangency question duplicated OWR-17290-001 and was removed; the distinct degenerate model was retained.