Landau-Level Multiplicities and Sign-Changing Spherical Interfaces

Definitions and mathematical statement

The supplied record contains two separate questions. Both require additional specifications before they define complete spectral problems.

For the Landau question, let $\Sigma$ denote a curve in the source's class of general finite $C^{1,1}$-smooth curves. Here $C^{1,1}$ means that local regular parametrizations are continuously differentiable with locally Lipschitz first derivative. The meaning of "finite," the ambient space, and the admissible endpoint and component configurations are not specified. Once the magnetic Hamiltonian, magnetic field, interaction strength, Hilbert space $\mathcal H$, and operator domain have been fixed, write $H_{\Sigma,\delta}$ for the resulting operator with a delta interaction supported on $\Sigma$. For an eigenvalue $\lambda$, its geometric multiplicity is \[ m_{\Sigma,\delta}(\lambda) :=\dim\ker\left(H_{\Sigma,\delta}-\lambda I_{\mathcal H}\right). \] The first requested task is to determine the multiplicities of the "perturbed Landau levels" for general such $\Sigma$. This terminology is unresolved: the record does not specify whether the target consists of the original Landau energies that remain eigenvalues after perturbation or eigenvalues shifted from those energies. No particular operator realization, coupling, energy normalization, or interpretation of this phrase is imposed here.

For the separate spherical-interface question, set \[ S^2:=\{x\in{\mathbb{R}}^3:\lVert x\rVert=1\}. \] The reconstructed statement concerns an interface $\Gamma\subset S^2$, written in the original extraction as the common boundary of a plus region and a minus region. Write $c\in{\mathbb{R}}$, $c<-1$, for the material contrast mentioned in the record. If $\Lambda(\Gamma,c)\subseteq{\mathbb{C}}$ denotes the eigenvalue-parameter set of the intended sign-changing elliptic spectral problem, the requested exclusion property is \[ \Lambda(\Gamma,c)\cap \left\{-\frac12+i\eta:\eta\in{\mathbb{R}}\setminus\{0\}\right\} =\varnothing. \] The second requested task is to characterize interfaces satisfying this property when the contrast is below $-1$. The symbol $\Lambda$ is only a placeholder: the differential equation or operator pencil, coefficients, interface conditions, admissible interface regularity, and function spaces are absent. The record also does not determine whether the characterization is for each fixed $c<-1$ or requires the property for every $c<-1$. The parameter $\eta=0$ is explicitly excluded from the requested prohibition.

Short English statement

The record asks two separate questions: determine multiplicities associated with Landau levels after adding a delta interaction on a general finite $C^{1,1}$ curve, and characterize spherical interfaces whose intended sign-changing elliptic problem has no spectral parameters $-1/2+i\eta$ with real $\eta\ne0$ at material contrast below $-1$. Essential operator specifications and some quantifiers are missing, so these are qualified statement-only drafts requiring mathematical review.

Sources

• UnsolvedMath local registry, permanent ID $30005614$, exact source problem code catalogue entry, title "Landau-Level Multiplicities and Sign-Changing Spherical Interfaces," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2023.
• Supplied report citation: Geometric Spectral Theory (2024), Oberwolfach Reports, pp. 2017--2106, DOI 10.4171/owr/2023/36. Original OWR report: https://doi.org/10.4171/owr/2023/36. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Behrndt--Holzmann--Lotoreichik--Raikov, "Landau delta interactions," https://doi.org/10.4171/JST/422. No fuller bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports reductions of multiplicity questions at perturbed Landau levels to Berezin--Toeplitz kernels, results for circles and generic cases, and an unresolved general finite $C^{1,1}$ interface problem. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft.