Dynamics of Noncommutative Solitons I: Spectral Theory and Dispersive Estimates

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.