On the Weyl function for complex Jacobi matrices

TL;DR

The work addresses representing the Weyl function for complex Jacobi matrices, including finite and semi-infinite cases, by connecting spectral questions to a discrete-time dynamical system with boundary controls. It advances the theory by employing Autonne-Takagi factorization to obtain a spectral representation of the finite-dynamics and then uses a discrete Fourier transform to translate time-domain responses into spectral data, yielding representations of $m(\lambda)$ and $m^N(\lambda)$ in terms of dynamic response vectors and a complex spectral measure. A key contribution is establishing complex-analytic Weyl-function formulas $m(\lambda)=-\sum_{t=0}^\infty z^t r_t$ and $m^N(\lambda)=-\sum_{t=0}^\infty z^t r^N_t$ for $\lambda$ in a region $D$, with $z$ tied to $\lambda$ by $\lambda=z+z^{-1}$, and tying these to a discrete measure $d\rho$ (and $d\rho^N$) that generalizes the real-case spectral measure. This framework extends complex moment problem techniques and provides a path to analyze the Weyl function via dynamic-response data, with potential implications for understanding spectral properties of complex Jacobi operators.

Abstract

We derive a new representation for the Weyl function associated with the complex Jacobi matrix in the finite and semi-infinite cases. In our approach we exploit connections to the discrete-time dynamical system associated with these matrices.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Theorem 1
• Definition 3
• Definition 4
• Remark 1
• Proposition 1
• proof