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This talk describes joint work with Roman Bessonov and Peter

Yuditskii. In the spectral theory of self-adjoint and unitary

operators in one dimension (such as Schrodinger, Dirac, and Jacobi

operators), a half-line operator is encoded by a Weyl function; for

whole-line operators, the reflectionless property is a

pseudocontinuation relation between the two half-line Weyl functions.

We develop the theory of reflectionless canonical systems with an

arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy

Theorem property. This generalizes, to an infinite gap setting, the

constructions of finite gap quasiperiodic (algebro-geometric)

solutions of stationary integrable hierarchies. Instead of theta

functions on a compact Riemann surface, the construction is based on

reproducing kernels of character-automorphic Hardy spaces in Widom

domains with respect to Martin measure. We also construct unitary

character-automorphic Fourier transforms which generalize the

Paley-Wiener theorem. Finally, we find the correct notion of almost

periodicity which holds in general for canonical system parameters in

Arov gauge, and we prove generically optimal results for almost

periodicity for Potapov-de Branges gauge, and Dirac operators.