Crossings or equality of the second and third eigenvalues in the half-integer flux double-well case

Determine whether, for the magnetic Schrödinger operator H_α = (−ih∇ − A_α)^2 + V with double radial potential wells V(x) = v(|x − x_ℓ|) + v(|x − x_r|), Aharonov–Bohm vector potential A_α(x) = αF(x − x_ℓ) + αF(x − x_r), and half-integer flux (i.e., α/h ∈ Z + 1/2 so e(α/h) = 1/2), the second and third eigenvalues satisfy λ_2(h,α) = λ_3(h,α) or strictly λ_2(h,α) < λ_3(h,α) in the semiclassical limit h → 0; that is, ascertain whether a level crossing or exact degeneracy occurs between λ_2 and λ_3 in this setting.

Background

The paper studies semiclassical tunneling in a two-well potential V(x) = v(|x − x_ℓ|) + v(|x − x_r|) under an Aharonov–Bohm vector potential A_α(x) = αF(x − x_ℓ) + αF(x − x_r), where F(x) is the Aharonov–Bohm field with a pole at the origin, and x_ℓ = (−L/2,0), x_r = (L/2,0), with L > 2σ. The wells are symmetric and v has a unique non-degenerate minimum at 0 as specified in condition (1.5).

For half-integer flux (e(α/h) = 1/2), Theorem 1.4 provides leading-order asymptotics for the first four eigenvalue gaps: it shows λ_2 − λ_1 at scale e^{-S(v,L)/h}, λ_4 − λ_3 at scale e^{-2S(v,L)/h}, and only an o(e^{-S(v,L)/h}) control for λ_3 − λ_2, leaving unresolved whether λ_2 and λ_3 are strictly separated or coincide. Subsection 6.4 discusses symmetry considerations that could lead to multiplicity, but a definitive resolution of the λ_2 versus λ_3 relation is not provided.