Dynamics of Noncommutative Solitons I: Spectral Theory and Dispersive Estimates

TL;DR

This paper studies the spectral properties and decay estimates of noncommutative solitons modeled by a Schrödinger equation with a special difference operator, introducing a novel technique involving orthogonal polynomials.

Contribution

It provides the first detailed spectral and dispersive analysis of noncommutative solitons using innovative generating function methods.

Findings

Proves optimal decay rate of $t^{-1}\,log^{-2}t$ for solutions.

Introduces a new technique using orthogonal polynomial generating functions.

Establishes foundational spectral results for noncommutative soliton dynamics.

Abstract

We consider the Schr\"odinger equation with a 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 prove pointwise in time decay estimates, with the optimal decay rate $t^{-1}\log^{-2}t$ generically. We use a novel technique involving generating functions of orthogonal polynomials to achieve this estimate.