Reflectionless canonical systems, I. Arov gauge and right limits

Szegő's Theorem for Canonical Systems: the Arov Gauge and a Sum Rule

We consider canonical systems and investigate the Szegő class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of the canonical system. This sum rule provides a spectral theory gem in the sense proposed by Barry Simon.

Reflectionless canonical systems, II. Almost periodicity and character-automorphic Fourier transforms

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 t…

无反射典范系统的拓扑性质

Topological properties of reflectionless canonical systems

本文研究无反射典范系统空间的拓扑性质，并指出 $\operatorname{PSL}(2,\mathbb R)$ 的自然作用是分析关键。

无反射 Dirac 算子与典范系统

Reflectionless Dirac operators and canonical systems

本文研究开集上无反射典范系统中半轴 m 函数的全纯延拓关系，重点分析 Dirac 算子与典范系统的相互作用，并提出更一般的抽象框架。

无反射算子与自守Herglotz函数

Reflectionless operators and automorphic Herglotz functions

本文研究开集上无反射的典型系统与Dirac算子，通过万有覆盖建立其与自守Herglotz函数的一一对应，并分析自守测度，揭示无反射算子空间的拓扑、极值点及有限间隙算子空间的嵌入关系。