Square minimizes the lowest magnetic Dirichlet eigenvalue among rectangles of fixed area

Prove that for every aspect ratio a > 0 and every constant magnetic field B ∈ ℝ, the lowest eigenvalue λ₁^B(Ω_{a,a^{-1}}) of the magnetic Dirichlet Laplacian (−i∇−A)² with Dirichlet boundary conditions on the rectangle Ω_{a,b} = (−a/2,a/2)×(−b/2,b/2), where the vector potential A generates the constant field B, satisfies λ₁^B(Ω_{a,a^{-1}}) ≥ λ₁^B(Ω_{1,1}).

Background

The paper studies spectral optimization of the lowest eigenvalue of the magnetic Dirichlet Laplacian in rectangles under a homogeneous magnetic field. For general domains, the magnetic Erdős–Faber–Krahn inequality ensures the disk minimizes the principal eigenvalue at fixed area, but explicit solutions for rectangles with magnetic fields are unavailable due to loss of separability and lack of suitable symmetrization for complex-valued functions.

The authors formulate a rectangle-specific isoperimetric conjecture asserting the square minimizes the principal magnetic Dirichlet eigenvalue among unit-area rectangles. They prove the conjecture for weak magnetic fields via local optimality (simplicity and perturbation) and quantitative bounds, but the conjecture remains open in full generality.