Abstract
We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. More precisely, we introduce the class of Krein–Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of Krein on the connection between the Stieltjes moment problem and Krein–Stieltjes strings and Kac on the connection between the Hamburger moment problem and 2times 2 canonical systems with Hamburger Hamiltonians.
Highlights
Let {sk}k≥0 be a sequence of real numbers
[11] that there is a one-to-one correspondence between spectral problems (1.4) and canonical systems. This entails that every Herglotz– Nevanlinna function can be identified with the Weyl–Titchmarsh function of a unique spectral problem (1.4), for this, the assumptions on the coefficients have to be relaxed to allow ω to be a real-valued distribution in Hl−oc1([0, L)) and υ to be a positive Borel measure on [0, L)
Suppose that {sk}k≥0 is a strictly positive sequence and that ρ is a solution to the Hamburger moment problem (1.1)
Summary
Let {sk}k≥0 be a sequence of real numbers. The classical Hamburger moment problem is to find a positive Borel measure ρ on R such that the numbers sk are its moments of order k, that is, such that sk = λk ρ(dλ), k ≥ 0. [11] that there is a one-to-one correspondence between spectral problems (1.4) and canonical systems This entails that every Herglotz– Nevanlinna function can be identified with the Weyl–Titchmarsh function of a unique spectral problem (1.4), for this, the assumptions on the coefficients have to be relaxed to allow ω to be a real-valued distribution in Hl−oc1([0, L)) and υ to be a positive Borel measure on [0, L). Our motivation to include the classical moment problem in the spectral theory of generalized indefinite strings is dictated by the study of infinite multi-peakon solutions of the Camassa–Holm equation (see [2,9,12] for the case of finitely many peakons). After these preparations, we proceed to introduce the class of Krein–Langer strings in Sect. The space Hl−oc1[0, L) is the topological dual of Hc1[0, L)
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