Existence of Hamiltonians with vanishing ratio J(r)/(κ_H(r/\log r)·\log r)

Decide whether there exists a Hamiltonian H (in limit circle case) for which the parameter-optimised upper bound J(r) from Romanov’s Theorem I becomes asymptotically negligible compared to the bound \kappa_H(r/\log r)·\log r, in the precise sense that liminf_{r→∞} J(r)/[\kappa_H(r/\log r)·\log r] = 0.

Background

Comparisons between the best bounds obtainable via Romanov’s discretisation (J(r)) and the explicit κ_H-based bound reveal at most logarithmic gaps, but it is unknown whether these gaps can collapse for some Hamiltonians.

A construction (or impossibility result) would delineate the sharpness of κ_H-based bounds and the limits of discretisation-based optimisation.