Essential Spectrum of Sign-Indefinite Transmission Laplacians

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^n$ be a bounded domain, where the dimension $n$ is not specified in the supplied record. The source describes two material regions $\Omega_+$ and $\Omega_-$, separated by an interface $\Sigma$, with coefficient \[ h(x)=\begin{cases} 1,&x\in\Omega_+,\\ -\mu,&x\in\Omega_-. \end{cases} \] The formal differential expression is \[ \mathcal L_\mu u=-\nabla\cdot(h\nabla u). \] The spectral questions concern the critical contrast $\mu=1$. The record does not specify the dimension, regularity of the domain and interface, exterior boundary conditions, transmission conditions, or operator domain. Thus this expression alone does not determine an operator whose spectrum can be studied. In what follows, $L$ denotes the intended self-adjoint transmission realization in $L^2(\Omega)$, once those missing choices have been supplied; no particular realization is selected here.

For such a realization, use the convention \[ \sigma_{\mathrm{ess}}(L)=\sigma(L)\setminus\sigma_{\mathrm{disc}}(L), \] where $\sigma_{\mathrm{disc}}(L)$ consists of isolated eigenvalues of finite multiplicity. The supplied questions are:

1. At $\mu=1$, can a configuration in the intended class have two distinct points in $\sigma_{\mathrm{ess}}(L)$? Can its essential spectrum contain a nonempty interval? The source does not specify the admissible geometric class or whether "interval" must have positive length. The stronger, positive-length interpretation would require \[ \exists\,a<b\quad [a,b]\subseteq\sigma_{\mathrm{ess}}(L). \] This interpretation is recorded as a possible intended target, not imposed as a supplied hypothesis.
2. If $\sigma_{\mathrm{ess}}(L)=\{\lambda_*\}$, determine the accumulation behavior and rates of discrete eigenvalues approaching $\lambda_*$. For any such sequence, counted with multiplicity, the relevant limit is \[ \lambda_k\in\sigma_{\mathrm{disc}}(L),\qquad \lambda_k\longrightarrow\lambda_*. \] Neither the existence of such a sequence nor a particular accumulation rate is asserted by the record. In particular, the singleton is not stipulated to be $\{0\}$ in the general question.
3. Develop boundary-triple methods for other relative positions of the material regions, described literally in the source as removing the assumption $\Omega_-\subseteq\Omega$, and for nonsmooth interfaces $\Sigma$. The literal inclusion is already inherent if $\Omega_-$ is a material subregion of $\Omega$. Whether the intended restriction is compact containment, separation from the exterior boundary, or another geometric condition is unresolved. No replacement hypothesis or class of nonsmooth interfaces is supplied.

The supplied dated literature assessment reports special smooth-interface results: compact resolvent for strictly convex interfaces and essential spectrum at zero for flat parts. It reports the broader questions on larger essential spectra, accumulation rates, other relative positions, and nonsmooth interfaces as unresolved in general. These are supplied curation claims, with their precise hypotheses absent from this record; they are not independently verified results or a new literature assessment.

Short English statement

At critical contrast $\mu=1$, the problem asks how large the essential spectrum of a self-adjoint sign-indefinite transmission Laplacian can be, how discrete eigenvalues approach a singleton essential spectrum, and how boundary-triple methods extend to other geometries and nonsmooth interfaces. The operator realization and geometric hypotheses need clarification before these questions become a fully specified mathematical problem.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004277$, exact source problem code catalogue entry, title Essential Spectrum of Sign-Indefinite Transmission Laplacians, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2019.
• Supplied 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 background identifies Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Cacciapuoti--Pankrashkin--Posilicano, "self-adjoint indefinite Laplacians" (supplied reference label), DOI 10.1007/s11854-019-0057-z. https://doi.org/10.1007/s11854-019-0057-z.
• Supplied literature source labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2019/47.
• Literature assessment date: 2026-08-22. The supplied assessment labels the problem open, reports the smooth-interface special cases described above, and regards the broader questions as unresolved in general. This is supplied dated curation derived from the OWR record, not independently verified; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired for self-contained publication. This supplied editorial label does not resolve the formulation issues identified here or constitute independent mathematical review.