Constructive synthesis of Hamiltonians with prescribed regularly varying growth

Develop a constructive procedure that, for any regularly varying function g satisfying \log r = o(g(r)) and g(r) = o(r), produces an explicit two-dimensional canonical system Hamiltonian H in limit circle case whose monodromy matrix satisfies \log(max_{|z|=r} ||W_H(z)||) \asymp g(r). Provide constructions covering both discrete (Hamburger) and continuous rotation cases, beyond presently known families.

Background

An existence theorem shows that, for any regularly varying g between \log r and r, there is a canonical system whose monodromy matrix has growth \asymp g. Explicit constructions are known for certain smoothly varying g (e.g., via Hamburger Hamiltonians with l_n ≍ 1/f(n) or continuous rotation using φ(t) inverse to r/f(r)), but no general constructive method is available.

A full constructive solution would enable systematic synthesis of models with targeted spectral density profiles and facilitate applications across related operator classes.