Structure of Noncommutative Solitons: Existence and Spectral Theory

TL;DR

This paper studies the spectral properties of a noncommutative soliton modeled by a second order difference operator, establishing existence, decay estimates, and spectral theory insights relevant to noncommutative field theories.

Contribution

It constructs a ground state soliton for a nonconstant coefficient difference operator and analyzes its decay and spectral properties, advancing understanding of noncommutative solitons.

Findings

Existence of a ground state soliton with specific decay properties

Derivation of $ ext{l}^{ ext{infty}}$ and $ ext{l}^{1}$ estimates

Identification of a quasi-exponential spatial decay rate

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 construct a ground state soliton for this equation and analyze its properties. In particular we arrive at $\ell^{\infty}$ and $\ell^{1}$ estimates as well as a quasi-exponential spatial decay rate.