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.