Structure of Noncommutative Solitons: Existence and Spectral Theory

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.