The Riemann problem, complete integrability and arithmetic applications

Description

This volume, "Seminar on the Riemann Problem, Complete Integrability and Arithmetic Applications", contains a series of lectures presented at a seminar of the same title given by D.and G. Chudnovsky and held in 1979-1980 at the Institute des Hautes Etudes Scientifiques (IHES) in Bures-sur-Yvette, France (1979),' and at Columbia University in the City of New York, U.S.A. The Seminar speakers examine different aspects of analytic and arithmetic problems arising in various ways from contemporary studies of the Riemann boundary value and monodromy problems. Particular subdivisions of the volume are the following: studies in spectral theory and completely integrable systems (inverse scattering method); the Riemann monodromy problem and statistical mechanics; Pade approximations associated with the Riemann boundary value problem and arithmetical applications to transcendental numbers.

We want to express our profound gratitude to the authors who contributed to this volume for their wonderful presentations at the Seminar and contributions to the diverse and fascinating subject, and for the preparation of manuscripts.

We want to thank the participants of the Seminar at IHES and Columbia University. Professor N. Kuiper (Director of IHES) made it possible for the Seminar to meet at IHES. Our special thanks go to colleagues M. J. Ablowitz, L. Bers, D. Bessis, H. Cornille, J. Frohlich, F. Giirsey, H. Jacguet, R. Jost, A. Neveu, and A. Voros for their invaluable discussions on the subject of the Seminar. The editors acknowledge with gratitude partial support extended to the editors by CNRS and CEN-Saclay in France and ONR and NSF in the United States.

We warmly thank F. Brown for her constant help during the preparation of this volume and K. March for typing the manuscript. We open the volume with an introduction in which we try to summarize the seemingly disconnected and various aspects of applications of the Riemann boundary value problem. The purpose of this is to allow immediate access, for students of the subjects as well as for teachers presenting special courses on the Riemann problem, to the contemporary research literature in this rapidly changing field-D,andG.Chudnovsky,

INTRODUCTION 1
1. STATISTICAL MECHANICS AND THE RIEMANN MONODROMY PROBLEM
1.1 B.M. McCoy and J.H.H. Perk, Continuous exponents of spin corellation functions of inhomogeneous layered Ising models 12
1.2 T. Miwa and M. Jimbo, Introduction to holonomic quantum fields 28
1.3 D.B. Abraham, Planar Ising ferromagnet: correlation functions and the inverse scattering method. 37
2. COMPLETELY INTEGRABLE SYSTEMS
2.1 D.V. Chudnovsky, Infinite component two-dimensional completely integrable systems of KdV type 57
2.2 D.V. Chudnovsky, Infinite component cr-models and instanton solutions 71
2.3 D.V. Chudnovsky, The representation of an arbitrary, two-dimensional completely integrable system as the common action of two commuting one-dimensional Hamiltonian
flows 8 5
2.4 J.P. Bourguignon, Self-duality of Yang-Mills fields and of gravitational instantons 95
2.5 R.C. Churchill, On proving the nonintegrability of a
Hamiltonian system 103
2.6 M.S. Berger, Classical solutions in nonlinear Euclidean field theory and complete integrability 123
2.7 D.V. Chudnovsky, G.V. Chudnovsky, Hamiltonian structure of isospectral deformation equations. Elliptic curve case 134
2.8 D.V. Chudnovsky, G.V. Chudnovsky, Quantum Hamiltonians associated with finite-dimensional Lie algebras and vactorized S-matrices 147
VI
2.9 D.V. Chudnovsky, G.V. Chudnovsky, A. NeveU, Classical and quantum operator nonlinear Schrodinger equation I 157
3. SPECTRAL PROBLEMS
3.1 G. Parisi, Trace identities for the Schrodinger operator and the WKB method 178
3.2 A. Voros, Zeta functions of the quartic (and homogeneous anharmonic) oscillators 184
3.3 L. Bers, On trace formula 209
3.4 D.V. Chudnovsky, G.V. Chudnovsky, Resolvent and trace identities in the one-dimensional 215
3.5 S. Aubry, The devil's staircase transformation in incommersurate lattices 221
4. THE PADE APPROXIMATION, THE RIEMANN BOUNDARY VALUE
PROBLEM AND ARITHMETIC APPLICATIONS
4.1 J. Nuttall, The convergence of Pade approximants and their generalizations 246
4.2 J.L. Gammel and J. Nuttall, Note on generalized Jacobi polynomials 25S
4.3 D.V. Chudnovsky, G.V. Chudnovsky, Multidimensional Hermite interpolation and Pade approximation 271
4.4 G.V. Chudnovsky, Hermite-Pade' approximations to exponential functions and elementary estimates of the measure of irrationality of Pi 299
4.5 G.V. Chudnovsky, Criteria of algebraic independence of several numbers. 323
4.6 K.H. Prendergast, Rational approximation for nonlinear ordinary differential equations 369

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