Extend the Demuth–Hansmann–Katriel inequality to 0<γ<1 for σ>d/2

Determine whether, for every space dimension d≥1, parameters 0<γ<1 (with γ≥1/2 when d=1), and σ>d/2, there exists a constant C_{γ,σ,d} such that for every complex-valued potential V∈L^{γ+d/2}(R^d) and every ħ>0, the discrete eigenvalues E_j of the Schrödinger operator H_{ħ,V}=−ħ^2Δ+V in L^2(R^d) satisfy the bound ∑_j |E_j|^{−σ} δ(E_j)^{γ+σ} ≤ C_{γ,σ,d} ħ^{−d} ∫_{R^d} |V(x)|^{γ+d/2} dx, where δ(E)=dist(E,[0,∞)).

Background

The Demuth–Hansmann–Katriel (DKH) inequality provides an upper bound on eigenvalue sums for Schrödinger operators with complex potentials, currently established for γ≥1 and σ>d/2. It interpolates the Frank–Laptev–Lieb–Seiringer cone-restricted inequality and reduces to the classical Lieb–Thirring bounds when the potential is real-valued.

The open question asks whether the same type of inequality extends to the regime 0<γ<1 (with the natural restriction γ≥1/2 in one dimension), maintaining the σ>d/2 threshold. An affirmative result would significantly broaden the applicability of DKH-type bounds to weaker integrability assumptions on V while retaining the same semiclassical scaling with respect to ħ.