Nevanlinna matrix estimates without regularity conditions
Jakob Reiffenstein
TL;DR
The paper addresses the order problem for the Nevanlinna matrix of half-line Jacobi operators by connecting it to the monodromy matrix of a discrete, Hamburger Hamiltonian canonical system. It develops universal lower bounds that rely only on dataset-derived quantities $b_j^{(s)}$ built from the Hamiltonian, and establishes flexible upper bounds that can be tuned via determinant controls on the Hamiltonian segments. The results show that the order $\rho$ of the Nevanlinna matrix is bounded below by the convergence exponent of natural sequences of Jacobi parameters and, under broad LC assumptions, can equal this exponent, thereby generalising Berezanskii's theorem with significantly relaxed regularity conditions. The framework also elucidates how summability of angle increments and convergence rates of angles influence both lower and upper bounds, yielding sharp predictions in a variety of decay regimes. Overall, the work provides robust, regularity-free tools to relate spectral density growth to explicit parametric data in both Hamiltonian and Jacobi settings, with implications for the LC/limit-point dichotomy and eigenvalue distribution.
Abstract
The Nevanlinna matrix of a half-line Jacobi operator coincides, up to multiplication with a constant matrix, with the monodromy matrix of an associated canonical system. This canonical system is discrete in a certain sense, and is determined by two sequences, called "lengths" and "angles". We derive new lower and upper estimates for the norm of the monodromy matrix in terms of the lengths and angles, without imposing any restrictions on these sequences. Returning to the Jacobi setting, we show that the order of the Nevanlinna matrix is always greater than or equal to the convergence exponent of the off-diagonal sequence of Jacobi parameters, and obtain a generalisation of a classical theorem of Berezanskii.