The Classical Moment Problem and Generalized Indefinite Strings

Abstract

We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, 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 M. G. Krein on the connection between the Stieltjes moment problem and Krein-Stieltjes strings and I. S. Kac on the connection between the Hamburger moment problem and 2 × 2 canonical systems with Hamburger Hamiltonians.

Heat kernels of the discrete Laguerre operators

“…Observe that the matrix ( 1.1 ) admits “the string factorization” (see, e.g., [ 3 , Appendix], [ 14 , § 13], [ 6 , §3]): where and Therefore, the Jacobi matrix ( 1.1 ) can be (at least formally) written as where and are the multiplication operators is the shift operator with the standard convention , and is the backward shift, . The representation ( 3.8 ) immediately implies for every .…”

“…The string factorization ( 3.5 ) implies the following Stieltjes continued fraction representation of the Weyl function (see, e.g., [ 14 , §13], [ 28 , 6 , § 3]): which converges locally uniformly in (this follows from the self-adjointness of , see, e.g., [ 3 , 6 , §3]). Taking into account ( 3.6 ) and ( 3.7 ) and noting that for all , we arrive at ( 3.14 ).…”

The Inverse Spectral Problem for Periodic Conservative Multi-peakon Solutions of the Camassa–Holm Equation

“…By means of using Stieltjes-type formulas (see [43] or [28], [31]) for the coefficients in the continued fraction expansion in (3.10), it is possible to recover solutions of the inverse Dirichlet spectral problem explicitly in terms of the spectral data. From this, one obtains the following fact which characterizes the subclass of P that gives rise to purely positive (negative) Dirichlet spectrum.…”

On the Absolutely Continuous Spectrum of Generalized Indefinite Strings

“…Spectral problems of this type are of interest for at least two reasons. Firstly, they constitute a canonical model for operators with simple spectrum; see [20,22]. Secondly, they are of relevance in connection with certain completely integrable nonlinear wave equations (most prominently, the Camassa-Holm equation [9]), for which these kinds of spectral problems arise as isospectral problems.…”