August Krueger, Technion

**Title:** Structure and Dynamics of Noncommutative Solitons: Spectral Theory and Dispersive Estimates

**Abstract:**

We consider the Schroedinger equation with a Hamiltonian given by a second order difference operator with nonconstant growing coefficients on the half line. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of string theory and noncommutative field theory. We prove pointwise in time decay estimates with the optimal decay rate generically. We use a novel technique involving generating functions of orthogonal polynomials to achieve these estimates. We construct a ground state soliton for this equation and analyze its properties. We completely determine the spectrum of the associated linearized Hamiltonian and prove the optimal rate for the associated time decay estimate. This work has been conducted in collaboration with Avy Soffer.