Dynamics of Noncommutative Solitons II: Spectral Theory, Dispersive Estimates and Stability

TL;DR

This paper analyzes the spectral properties and decay estimates of a matrix Schrödinger operator arising in noncommutative soliton dynamics, introducing new techniques for spectral analysis and decay rate determination.

Contribution

It provides a complete spectral characterization and optimal decay rates for the linearized operator around noncommutative solitons, using innovative methods involving orthogonal polynomials.

Findings

Spectrum of the linearized Hamiltonian is fully determined.

Optimal decay rate of t^{-1} log^{-2} t established.

Novel technique using generating functions of orthogonal polynomials introduced.

Abstract

We consider the Schr\"odinger equation with a (matrix) Hamiltonian given by a second order difference operator with nonconstant growing coefficients, on the half one dimensional lattice. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of noncommutative field theory. We completely determine the spectrum of the Hamiltonian linearized around a ground state soliton and prove the optimal decay rate of $t^{-1}\log^{-2}t$ for the associated time decay estimate. We use a novel technique involving generating functions of orthogonal polynomials to achieve this estimate.