Colloquium
3:00 p.m., Wednesday (January 26, 2005)
WEST MALL ANNEX 110
Michael Goldberg
Caltech
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 timeindependent potential, and
if so, under what conditions?
In the threedimensional setting, the following criteria are known
to be sufficient: An integrability condition regulating the
singularities and decay of the potential, and a zeroenergy 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).
