The Metaplectic Group, Harmonic Oscillators, and Singular Perturbation
Series
John Stalker
Princeton University
Abstract:
I will talk about two closely related areas of mathematics,
one quite
beautiful, the other rather ugly. The beautiful
part, which is due to
many people, most of whom are long dead, is the connection
between the
classical mechanics of harmonic oscillators, the corresponding
quantum
systems, the Heisenberg group, the inhomogeneous metaplectic
group,
the Schwarz class of functions and tempered distributions,
and the
transformation law for theta functions. That sounds
like rather a
lot, and I may have to skip a few of the more interesting
digressions,
but I hope to get through the essentials of all of it.
All of this material
is well-known, but there doesn't seems to a be a single
source that puts
all the connections together. The ugly part of
the talk, which I will
try to keep as brief as possible, is due to me.
It uses the machinery
described in the first part of the talk to examine a
particular singular
pertubation problem in quantum mechanics. The problem
is fairly
natural, determining the behavior of transition amplitudes
for the
harmonic oscillator as functions of the spring constant,
but the
answer is quite ugly. If time permits I will deliver
a long,
distempered diatribe against perturbation theory as used
and abused in
quantum mechanics.
Room and Date:
Math Tower 154
Friday 4:30pm, October 29, 1999 at Ohio State University
Accomodation: Ramada University Hotel, October
29-31
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