Dispersive Bounds for the Schrodinger Equation with Almost Critical Potentials

Solutions to the Schrodinger equation are constrained in their evolution by several families of inequalities, among them the dispersive bounds. These typically control the L^{p'} norm of a solution at time t in terms of the L^p norm of the initial data and a polynomially decaying factor of the time elapsed. The familiar law of conservation of mass is expressed here in the special case p = 2. In this talk I will address the question: Are the dispersive bounds still valid in the presence of a time-independent potential, and if so, under what conditions?

In the three-dimensional setting, the following criteria are known to be sufficient: An integrability condition regulating the singularities and decay of the potential, and a zero-energy condition on the associated Hamiltonian. It is not necessary to assume that the potential posesses any additional regularity, positivity, or smallness. I will sketch a proof of this theorem, and describe the best known results in other dimensions.

Refreshments will be served at 2:45 p.m. in WMAX, Room 101 (the PIMS library).