Higher-dimensional violation of the ħ^{-d} locality bound

Construct, for each dimension d≥2 and parameters γ>0 and 0≤σ≤d/2, a complex-valued potential V∈L^{γ+d/2}(R^d) such that for the discrete eigenvalues E_j of the Schrödinger operator H_{ħ,V}=−ħ^2Δ+V in L^2(R^d) one has ∑_j |E_j|^{−σ} δ(E_j)^{γ+σ} ≫ ħ^{−d} as ħ→0, where δ(E)=dist(E,[0,∞)).

Background

In one dimension there are explicit examples of complex potentials V that produce eigenvalue sums growing strictly faster than the semiclassical ħ^{-d} scale, demonstrating a violation of locality: for instance, lower bounds of order ħ^{-1}(log(1/ħ)) or even ħ^{-2+2σ+ε} depending on σ. These exceed the semiclassical scaling predicted by locality-based bounds.

The open problem asks for analogous constructions in higher dimensions d≥2, within the natural integrability class L^{γ+d/2}(R^d), to show that the nonlocal growth persists and exceeds ħ^{-d} in the stated parameter range 0≤σ≤d/2.