Strong resolvent convergence for general non-resonant scaled potentials (without non-negativity)

Establish that, for any non-resonant potential V in C-infinity on the closed half-line with compact support (i.e., V in C^∞(overline{R_+}) and supp V ⊂ [0,a]) used to define the scaled boundary-distance-dependent potential V_ε(x) = ε^{-2} V(dist(x,Σ)/ε) on a bounded smooth domain Ω with boundary Σ, the family of Dirichlet Schrödinger operators A_ε = -Δ + V_ε with domain H^1(Ω) ∩ H^2(Ω) converges in the strong resolvent sense to the Dirichlet Laplacian A_0 = -Δ with domain H^1(Ω) ∩ H^2(Ω) as ε → 0, without assuming V ≥ 0. Here "non-resonant" means that the initial-value problem on R_+, -ψ'' + Vψ = 0 with ψ(0) = 0, has no bounded nontrivial solution.

Background

The paper analyzes Schrödinger operators on bounded smooth domains Ω with Dirichlet boundary conditions and potentials that depend only on the distance to the boundary Σ. The scaled potentials are of the form V_ε(x) = ε^{-2} V(dist(x,Σ)/ε), where V is defined on the half-line with compact support. The limit behavior depends on whether V is resonant or non-resonant in a one-dimensional sense.

For resonant V, the authors prove strong resolvent convergence to a Robin Laplacian with coefficient determined by the mean curvature of the boundary. For non-resonant V, they prove strong resolvent convergence to the Dirichlet Laplacian under the additional assumption V ≥ 0. They conjecture that this sign restriction is unnecessary and that the same Dirichlet limit should hold for all non-resonant V, but they do not provide a proof.

Later in the text (Main results subsection following Theorem [the non-resonant case]), they reiterate this point: by analogy with the one-dimensional case, they expect the strong resolvent convergence to the Dirichlet Laplacian for general non-resonant potentials and explicitly state that they have not been able to prove it.