Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula

We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum ${\textsf{E}}$ and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as ${\textsf{E}}$-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them “MCMV.” Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.

Real-valued algebro-geometric solutions of the Camassa–Holm hierarchy

We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa–Holm (CH) hierarchy and describe the associated isospectral torus. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint Hamiltonian systems. In particular, we rely on Weyl–Titchmarsh theory for singular (canonical) Hamiltonian systems. We also briefly discuss real-valued algebro-geometric solutions with a cusp behaviour. While we focus primarily on the case of stationary algebro-geometric CH solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.