Periodic GMP Matrices

TL;DR

This paper introduces GMP matrices, a new class generalizing Jacobi matrices related to the Moment Problem, and demonstrates their spectral properties and parametrization of almost periodic finite gap Jacobi matrices.

Contribution

It defines GMP matrices, extends spectral theory results from Jacobi matrices to this new class, and establishes a 'magic formula' analogue.

Findings

GMP matrices generalize Jacobi matrices related to the Moment Problem.

Parametrization of almost periodic finite gap Jacobi matrices using GMP matrices.

Established spectral theory results and a 'magic formula' for GMP matrices.

Abstract

We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to the strong Moment Problem. This class allows us to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover, due to their structural similarity we can carry over numerous results from the direct and inverse spectral theory of periodic Jacobi matrices to the class of periodic GMP matrices. In particular, we prove an analogue of the remarkable "magic formula" for this new class.