Structure of Noncommutative Solitons: Existence and Spectral Theory

We consider the Schrödinger 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 $${\ell^{\infty}}$$^ℓ∞ and $${\ell^{1}}$$^ℓ1 estimates as well as a quasi-exponential spatial decay rate.