Abstract: Solutions to the Schrödinger 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.