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We establish a sharp upper estimate for the order of a canonical system in terms of the Hamiltonian. This upper estimate becomes an equality in the case of Krein strings. As an application we prove a conjecture of Valent about the order of a certain class of Jacobi matrices with polynomial coefficients.

We study spectral properties of two-dimensional canonical systems y ′ (t) = zJH(t)y(t), t ∈ [a, b), where the Hamiltonian H is locally integrable on [a, b), positive semidefinite, and Weyl's limit point case takes place at b. We answer the following questions explicitly in terms of H:Is the spectrum of the associated selfadjoint operator discrete ?If it is discrete, what is its asymptotic distribution ?Here asymptotic distribution means summability and limit superior conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this correponds to convergence class and type w.r.t. proximate orders having order larger than 1. It is a surprising fact that these properties depend only on the diagonal entries of H. In 1968 L.de Branges posed the following question as a fundamental problem:Which Hamiltonians are the structure Hamiltonian of some de Branges space ?We give a complete and explicit answer.AMS MSC 2010: 37J05, 34L20, 45P05, 46E22

Abstract. Let µ be a measure on the real line R such that R dµ(t) 1+t 2 < ∞ and let a > 0. Assume that the norms f L 2 (R) and f L 2 (µ) are comparable for functions f in the Paley-Wiener space PWa and that PWa is dense in L 2 (µ). We reconstruct the canonical Hamiltonian system JX ′ = zHX such that µ is the spectral measure for this system.

We investigate the order ρ of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for ρ which is explicit in terms of the parameters of the canonical system associated with the moment sequence via its three-term recurrence. Under a weak regularity assumption this estimate coincides with a lower estimate, and hence ρ becomes computable. Dropping the regularity assumption leads to examples where upper and lower bounds do not coincide and differ from the order. In particular we provide examples for which the order is different from its lower estimate due to M.S.Livšic.