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Faber–Krahn-type inequality for quantum dot Dirac operators in two dimensions

Prove that for every bounded C^2 planar domain Ω and every disk D with |D|=|Ω|, if Ω is not a disk then the first nonnegative eigenvalue λ_Ω(θ) of the two-dimensional quantum dot Dirac operator D_θ with mass m≥0, defined by D_θ=(-i σ·∇+m σ_3) on L^2(Ω)^2 with boundary condition φ=(cosθ σ·τ+sinθ σ_3)φ on ∂Ω, satisfies λ_Ω(θ)>λ_D(θ) for all θ∈(-π/2,π/2).

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Background

The paper studies spectral shape optimization for two-dimensional quantum dot Dirac operators D_θ with mass m and boundary condition φ=(cosθ σ·τ+sinθ σ3)φ on ∂Ω. For each θ, the first nonnegative eigenvalue is λΩ(θ).

Motivated by conjectures in three dimensions for generalized MIT bag models and numerical evidence in the literature, the authors formulate the two-dimensional analogue as a Faber–Krahn-type statement asserting that disks minimize λ_Ω(θ) among domains of given area, with strict inequality unless Ω is a disk.

References

Conjecture. Assume that m≥0. Let Ω⊂R2 be a bounded domain with C2 boundary and let D⊂R2 be a disk with the same area as Ω. If Ω is not a disk, then λΩ(θ)> λ{D}(θ) for all θ∈(-π/2,π/2).

— A connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians in the context of shape optimization problems (2507.18698 - Duran et al., 24 Jul 2025) in Conjecture 1.1, Introduction – Subsection “The shape optimization problem”