The Classical Moment Problem and Generalized Indefinite Strings

TL;DR

This paper establishes a novel spectral theory framework linking the classical Hamburger moment problem to generalized indefinite strings, expanding the mathematical understanding of moment problems through Krein-Langer strings.

Contribution

It introduces Krein-Langer strings and demonstrates a bijective correspondence with moment sequences, bridging classical moment problems and spectral theory in a new way.

Findings

Established a bijective correspondence between moment sequences and Krein-Langer strings.

Extended spectral theory to include the classical Hamburger moment problem.

Connected classical moment problems with 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 2x2 canonical systems with Hamburger Hamiltonians.