**Memoirs of the American Mathematical Society**

1996;
111 pp;
Softcover

MSC: Primary 58;
Secondary 14; 35

Print ISBN: 978-0-8218-0460-5

Product Code: MEMO/122/581

List Price: $45.00

AMS Member Price: $27.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-0166-5
Product Code: MEMO/122/581.E**

List Price: $45.00

AMS Member Price: $27.00

MAA Member Price: $40.50

# Integrable Systems and Riemann Surfaces of Infinite Genus

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*Martin U. Schmidt*

This memoir develops the spectral theory of the Lax
operators of nonlinear Schrödinger-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.

#### Table of Contents

# Table of Contents

## Integrable Systems and Riemann Surfaces of Infinite Genus

- Contents vii8 free
- Introduction 110 free
- 1 An asymptotic expansion 716 free
- 2 The Riemann surface 1726
- 3 The dual eigen bundle 2332
- 4 The Riemann-Roch Theorem 3140
- 5 The Jacobian variety 4150
- 6 Darboux coordinates 5362
- 7 The tangent space of the Jacobian variety 6372
- 8 A reality condition 7382
- 9 The singular case 8998
- A: Borel summability 99108
- B: Another reality condition 104113
- Bibliography 109118