One of the most powerful strategies
for understanding solutions of linear partial differential equations
is to analyse the fundamental solution as precisely as possible.
In this lecture I will consider the fundamental solution of the
time-dependent Schrödinger equation, usually known as the
"propagator".
In contrast to the situation with
the heat or wave equations, where the fundamental solutions have
been well-understood for decades, the propagator has only recently
been precisely analysed, and then only in certain special geometrical
situations. In this lecture, drawing on some joint work of mine
with Wunsch, and with Tao and Wunsch, I will describe some properties
of the propagator when the domain is an asymptotically conic,
nontrapping manifold (a generalization of Euclidean space), and
give some applications.
