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We develop a theory of regularity for Dirac operators with uniformly locally square-integrable operator data. This is motivated by Stahl--Totik regularity for orthogonal polynomials and by recent developments for continuum Schr\"odinger operators, but contains significant new phenomena. We prove that the symmetric Martin function at $\infty$ for the complement of the essential spectrum has the two-term asymptotic expansion $\Im \left( z - \frac{b}{2 z}\right) + o(\frac 1z)$ as $z \to i \infty$, which is seen as a thickness statement for the essential spectrum. The constant $b$ plays the role of a renormalized Robin constant and enters a universal inequality involving the lower average $L^2$-norm of the operator data. However, we show that regularity of Dirac operators is not precisely characterized by a single scalar equality involving $b$ and is instead characterized by a family of equalities. This work also contains a sharp Combes--Thomas estimate (root asymptotics of eigensolutions), a study of zero counting measures, and applications to ergodic and decaying operator data., Comment: 34 pages

We develop a comprehensive 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 for canonical system parameters in Arov gauge, and we prove generically optimal results for almost periodicity for Potapov-de Branges gauge, and Dirac operators., Comment: 57 pages

In spectral theory, $j$-monotonic families of $2\times 2$ matrix functions appear as transfer matrices of many one-dimensional operators. We present a general theory of such families, in the perspective of canonical systems in Arov gauge. This system resembles a continuum version of the Schur algorithm, and allows to restore an arbitrary Schur function along the flow of associated boundary values at infinity. In addition to results in Arov gauge, this provides a gauge-independent perspective on the Krein-de Branges formula and the reflectionless property of right limits on the absolutely continuous spectrum. This work has applications to inverse spectral problems which have better behavior with respect to a normalization at an internal point of the resolvent domain., Comment: 26 pages