Growth estimates for Nevanlinna matrices of order larger than one half

Our objects of study are two-dimensional canonical systems that arise from indeterminate Hamburger moment problems and associated half-line Jacobi operators in limit circle case. The monodromy matrix of such a system coincides, up to a permutation of its entries, with the Nevanlinna matrix of the associated moment problem. Its growth relates to the density of eigenvalues of self-adjoint realisations of the system and the Jacobi operator, respectively. The order of the Nevanlinna matrix is known to be at most 1. In the case of "large" order, meaning order greater than one half, determining the growth of the monodromy matrix is known to be harder than for "small" order, i.e., order less than one half. As our main result, we establish an explicit new lower bound for data featuring a certain kind of monotonicity, which correctly describes the growth in the case of large order. Moreover, we compute the order of the Nevanlinna matrix of a limit circle Jacobi matrix with two-term power asymptotics in a critical case.