On Matrix Valued Schr\"odinger Operators on the Discrete Real Line: Resolvent Boundary Values, Limiting Absorption Principle, H\"older Regularity and Dispersive Estimates

TL;DR

This paper develops explicit resolvent representations, proves H"older continuity of boundary values, and establishes dispersive estimates for matrix-valued discrete Schr"odinger operators on the line, advancing spectral and time evolution analysis.

Contribution

It introduces an explicit Wronskian-based resolvent kernel, improves boundary value regularity results, and derives dispersive estimates using a novel one-dimensional explicit approach.

Findings

Explicit resolvent kernel representation derived

H"older continuity of resolvent boundary values proved

Dispersive estimates for time evolution established

Abstract

This work establishes new results on spectral theory and time evolution for matrix-valued discrete Schr\"odinger operators on the space of square-summable matrix sequences. The matrix-valued formalism is employed to streamline notation, offering a more elegant alternative to the equivalent vector-valued framework with matrix potentials. Our main contributions are threefold: first, we derive an explicit Wronskian-based representation for the resolvent's integral kernel; second, we prove H\"older continuity for the resolvent's boundary values; and third, we establish dispersive estimates for the time evolution. Our approach begins with the construction of Jost solutions using Volterra equations and the transmutation operator, leading to proofs of their H\"older regularity and bounds in Wiener algebra norms. From these solutions, we obtain the explicit kernel representation. This explicit characterization - a distinctive feature of the one-dimensional setting - enables the direct computation of the resolvent's boundary values. We subsequently establish a limiting absorption principle and demonstrate the H\"older continuity of these boundary values, achieving an improvement over classical results obtained through abstract higher-dimensional methods. Finally, this detailed resolvent characterization is leveraged to prove the dispersive estimates for the time evolution of the system.