Sharpness of the exponent in the radial eigenvalue sum bound; counterexample for β < 1

Construct a counterexample showing that, for Schrödinger operators H = −Δ + V on L^2(R^d) with complex radial potentials V ∈ L^q(R^d) (q ∈ ((d+1)/2, d)) and any p > ((d−1)q)/(d−q), the strengthened inequality (∑_j δ(z_j)^β |z_j|^{β p (1 − d/(2q)) − 1})^{q/(β p)} ≤ C_{p,q,β} ∫_{R^d} |V|^q dx fails for some β < 1, thereby establishing that the exponent q/p in Theorem 1 cannot be increased.

Background

The main theorem provides improved bounds for sums of eigenvalues of Schrödinger operators with complex radial potentials, with a nonlocal exponent q/p < 1. The authors conjecture that this exponent is optimal and cannot be increased.

They explicitly pose the problem of finding a counterexample to any attempted strengthening that would replace the accumulation exponent by β < 1, which would demonstrate sharpness of their bound.