Scattering theory for difference equations with operator coefficients

TL;DR

This work extends stationary scattering theory to second-order difference equations with operator-valued coefficients by introducing a block Jacobi framework on $\mathcal{H}=\ell^2(\mathbb Z,H)$. Under moment-type perturbation conditions, it establishes the existence and precise asymptotics of Jost solutions $U^{\pm}(z)$, proves that they form a fundamental system with a well-behaved Wronskian, and constructs the transfer and scattering matrices with continuity properties on the unit circle. The authors show that the essential spectrum remains the interval $[-2,2]$ under compact/trace-class perturbations, while the absolutely continuous spectrum is preserved under Kato–Rosenblum-type results; they also analyze eigenvalue accumulation, offering non-accumulation results under higher moment or closed-range Wronskian assumptions and providing quantitative bounds via perturbation determinants when only trace-class perturbations are present. Together, these results generalize discrete scattering theory to operator-valued coefficients, enabling spectral and scattering analyses in quantum lattice systems with internal degrees of freedom and potentially infinite-dimensional state spaces.

Abstract

We consider a second order difference equation with operator-valued coefficients. More precisely, we study either compact or trace class perturbations of the discrete Laplacian in the Hilbert space of bi-infinite square-summable sequence with entries in a fixed Hilbert space. We discuss its continuous and discrete spectrum, as well as properties of the associated scattering matrix.

Theorems & Definitions (93)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Definition 2.1
• Lemma 2.2
• proof