Contemporary Mathematics

Volume 458, 2008

Percy Deift at integer times

Carlos Tomei

To Percy, with deep affection, admiration and gratitude.

ABSTRACT.

This is the first of two lectures dedicated to a description of some

of Percy's mathematical achievements.

1. Introduction

During the meeting arranged to honor Percy's 60th birthday, two lectures were

dedicated to a description of (part of) his work. This text concentrates on Percy's

contributions on scattering theory on the line and integrable systems up to the

early nineties. On the other hand, this is not a short history of either subject.

The seventies and eighties were very active times, and I had to limit myself to few

examples, emphasizing aspects which are relevant to Percy's current activities.

In retrospect, it is remarkable how these areas are intertwined. Scattering

theory provide action-angle variables to infinite dimensional evolutions, Riemann-

Hilbert problems come up in the solution of differential equations by factorization

formulas. Schrodinger operators and tridiagonal matrices follow parallel concep-

tual tracks and band matrices indicate that infinitely many integrals for certain

non-linear PDE's are not enough to assert complete integrability. Conserved quan-

tities arise not only from physics, but from numerical analysis: whatever is being

approximated by an iterative algorithm may not change along the computation.

As a student of Percy, I frequently had the impression that things moved faster

that I could follow. The feeling did not change.

2. Direct and inverse scattering

Percy's contributions to the scattering theory of operators on the line con-

centrate on two texts. Inverse Scattering on the Line (ISL,

[6]),

written with

E.

Trubowitz, presents a full account of the direct and inverse scattering theory for

the Schrodinger equation on the line with real potential,

L

2

u

=

D

2

u

+

q(x)u,

where

Du

=

-iu',

1

(1

+

x

2

)q(x)dx

oo.

2000 Mathematics Subject Classification:

Primary 34125, 35Q15, 70H06; Secondary 35Q58

Key words and phrases.

Scattering theory, integrable systems.

The author is thankful for the many suggestions presented by a referee. This work is sup-

ported by CNPq and Faperj.

http://dx.doi.org/10.1090/conm/458/08926