Roman Bessonov and I uploaded a new paper titled “de Branges canonical systems with finite logarithmic integral” to arxiv today. This is a continuation of an earlier manuscript, see also the previous topic on “Szego theorem on the real line and Krein strings”. In the current version, we completed the project of describing the measures on the line with finite logarithmic integral. Assume is Poisson-finite measure, i.e., that

We define its logarithmic integral as

Existence of logarithmic integral, i.e., condition

plays a role in the theory of Gaussian stationary stochastic processes: it holds if and only if the future of the process with spectral measure can not be predicted by its past.

Every Poisson-finite measure gives rise to a function in Herglotz-Nevanlinna class, i.e., the class of functions analytic in with non-negative imaginary part, by a Herglotz formula

where . The de Branges theory of Hilbert spaces of functions of exponential type provides a bijection between Herglotz-Nevanlinna class and the class of all canonical Hamiltonian systems. Canonical Hamiltonian system can be written as the following Cauchy problem

where Hamiltonian is nonnegative locally summable matrix-function on . In short, the Weyl-Titchmarsh theory for canonical systems provides for each and the converse is true by de Branges theory. In the paper, we characterize all for which the logarithmic integral of exists. Consider for which . Then, define the grid of points by the formula

and consider the quantity

Our main theorem, which is stated below, is a natural generalization of the Szego theorem in the theory of polynomials orthogonal on the unit circle.

**THEOREM. The measure has finite logarithmic integral if and only if for generated by .**

We quantify it by the sharp two-sided estimate. The sum in the definition of can be written in the form reminiscent of the matrix Muckenhoupt condition but we do not understand yet how the problem is connected to Muckenhoupt classes.

There are multiple applications of our theory to scattering problems for Dirac and wave equations.