This memoir develops the spectral theory of the Lax
operators of nonlinear SchrÃ¶dinger-like partial differential
equations with periodic boundary conditions. Their spectral curves,
i.e., the common spectrum with the periodic shifts, are generically
Riemann surfaces of infinite genus. The points corresponding to infinite
energy are added. The resulting spaces are no longer Riemann surfaces in
the usual sense, but they are quite similar to compact Riemann surfaces.
In fact, some of the basic tools of the theory of compact
Riemann surfaces are generalized to these spectral curves and illuminate
the structure of complete integrability:

The eigen bundles define holomorphic line bundles on
the spectral curves, which completely determine the potentials.

These line bundles may be described by divisors of the same
degree as the genus, and these divisors give rise to Darboux
coordinates.

With the help of a Riemann-Roch Theorem, the isospectral sets
(the sets of all potentials corresponding to the same spectral curve)
may be identified with open dense subsets of the Jacobian varieties.

The real parts of the isospectral sets are infinite dimensional
tori, and the group action solves the corresponding nonlinear partial
differential equations.

Deformations of the spectral curves are in one to one
correspondence with holomorphic forms.

Serre Duality reproduces the symplectic form.

Readership

Graduate students, research mathematicians,
and physicists interested in global analysis and analysis
on manifolds.