Bounds on Order of Indeterminate Moment Sequences

Abstract

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.

Estimates for the order of Nevanlinna matrices and a Berezanskii-type theorem

“…The estimate from theorem 4.1 is incomparable with the one obtained recently in [31]; in some cases it is better, and in some others it is worse, cf. proposition 4.5 and example 4.6.…”

“…Example 4.8. We revisit the Hamiltonians constructed in proposition 4.5 (so to make sure that order cannot be computed already from [31]), and consider the associated Jacobi matrices. Let h be a decreasing sequence with (4.3) which has the property that ((h n )/(h n+1 )) 1.…”

“…The proof is obtained by evaluating the upper estimate theorem 4.1 with help of [16], and combining this with a lower estimate from [31].…”

Density of the spectrum of Jacobi matrices with power asymptotics

“…Now we write the Jacobi matrix as a Hamburger Hamiltonian of a canonical system, cf. [18] or [13] for details about this reformulation. We denote by (l n ) ∞ n=1 and (φ n ) ∞ n=1 the sequences of lengths and angles of the corresponding Hamburger Hamiltonian.…”

“…With the notation from [18], the lengths and angle-differences are regularly distributed. Moreover, we have ∆ l = −2γ and ∆ φ = β + 2γ, both expressions exist as a limit and ∆ l + ∆ φ = β.…”