Optimality of parameter choices in Romanov’s discretisation bound

Determine whether the deterministic, r-dependent choice of rotation and distortion parameters used to produce the upper bound in the partition-only version of Romanov’s Theorem I is asymptotically optimal among all parameter choices in Romanov’s discretisation approach. Equivalently, decide whether, for every Hamburger (limit circle, det H ≡ 0) Hamiltonian, \log(max_{|z|=r} ||W_H(z)||) \asymp J(r), where J(r) denotes the infimum of upper bounds obtained from Romanov’s Theorem I over all admissible parameter sets.

Background

Romanov’s Theorem I bounds the monodromy growth by discretising the interval and choosing rotation and distortion parameters. A specific r-dependent choice leads to near-optimal bounds up to logarithmic factors, but it is unknown whether better choices can systematically improve the asymptotics.

Establishing optimality would clarify whether further parameter tuning can strengthen upper bounds or whether the current recipe is essentially best possible for broad classes of Hamiltonians.