Effective Hamiltonian for critical spectral accumulation

Determine an effective Hamiltonian for the critical Landau–Dirac operator with shell interactions that accounts for the interaction between the Landau–Dirac levels and the additional essential-spectrum interval I_Σ, thereby establishing the analogue of the non-critical reduction used to control eigenvalue accumulation near the Landau–Dirac levels.

Background

In the non-critical regime, the eigenvalue-counting problem near each Landau–Dirac level is reduced to a Toeplitz-type effective Hamiltonian involving the compact resolvent difference Υ=(D_{ε,τ}−ζ){-1}−(D_0−ζ){-1}. This compactness enables the authors to obtain operator inequalities and an O(1) control of the counting functions.

In the critical regime ε²−τ²=4, the corresponding effective Hamiltonian is no longer compact because it involves an unbounded operator on the supporting curve Σ. Consequently, the reduction underlying the non-critical analysis cannot be applied. The unresolved problem is to identify an effective Hamiltonian of a different type that captures the coupling between the Landau–Dirac levels and the additional essential-spectrum interval I_Σ=ran(−mτ/ε), with the broader aim of determining the critical-case spectral accumulation.

References

In the critical case, however, the effective Hamiltonian is no longer compact as it involves an unbounded operator on $\Sigma$ (see Lemma \ref{L BS}). This prevents us from obtaining the analogue of Lemma \ref{le1}, where the $O(1)$ bound crucially depends on compactness. One might expect an effective Hamiltonian of a different nature that accounts for an interaction between the Landau--Dirac levels and $I_\Sigma$. We do not address this issue in the current paper.

— The Landau-Dirac operator with shell interactions: self-adjointness and clustering  (2608.20665 - Benhellal et al., 21 Aug 2026) in Introduction, paragraph beginning “Our approach to addressing the clustering of eigenvalues”