Sharpness and scope of Frank–Sabin-type nonlocal Lieb–Thirring bounds

Ascertain, for each space dimension d≥1 and for the parameter triples (α,β,γ) listed in items (a)–(f), whether the nonlocal inequality ∑_j |E_j|^{α} (δ(E_j)/|E_j|)^{β} ≤ C_{α,β,γ,d} (ħ^{−d} ∫_{R^d} |V(x)|^{γ+d/2} dx)^{α/γ} is sharp up to a factor of ħ^{ε} for every ε>0; and determine whether the parameter region in which such inequalities hold can be expanded.

Background

Frank and Sabin, and subsequently Frank, established families of nonlocal eigenvalue bounds for Schrödinger operators with complex potentials that retain scale invariance but lose locality (since α/γ>1). These include several parameter regimes (a)–(f), some with truncations on the eigenvalue ranges.

While specific one-dimensional cases are known to be sharp up to logarithmic factors, the general sharpness (up to powers of ħ) and the maximal parameter range for which such inequalities hold remain unclear. Establishing sharpness would clarify the extent of the loss of locality and guide potential improvements.