J¨ urgen Moser Hamiltonian Systems and Spectral Theory Integrable Moscow Izhevsk 2003 Published by Regular and Chaotic Dynamics, Moscow-Izhevsk Universitetskaya, 1, Izhevsk, Russia, 426034 Phone: (7–3412) 50–02–95 Fax: (7–3412) 50–02–95 E-mail: borisov@rcd.ru Acknowledgement. <...> Moser, J¨ urgen INTEGRABLE HAMILTONIAN SYSTEMS AND SPECTRAL THEORY c 2003 by Institute of Computer Science, Moscow-Izhevsk 2003 by Regular and Chaotic Dynamics, Moscow-Izhevsk c All rights reserved. <...> The Scattering Problem Associated with the Equation of Kac and Van Moerbeke . . . . . . . . . 60 Various Aspects of Integrable Hamiltonian Systems . . . . 65 § 1. <...> Integrals for the Geodesic Flow on the Ellipsoid . . 140 Isospectral Deformation . . . . . 143 Interpretation of the Eigenvalues and the Frame of L . 145 Joachimsthal’s Integral . . . . . 147 § 4. <...> The Discrete Version of the Dynamics of a Rigid Body . . 246 1.1. <...> Explicit Formulas for the Discrete Dynamics of the 3-Dimensional Rigid Body . . . . . 261 § 2. <...> Discrete Version of the Neumann System and the Heisenherg Chain With Classical Spins . . . . . 266 § 3. <...> Kl. IIa, 1953, 4) Singular perturbation of eigenvalue problems for linear differential equations of even order. <...> Nauk, 38, 1983, 149–193. 60) Finitely many mass points on the line under the influence of an exponential potential — An integrable system. <...> Rational and elliptic solutions of the Korteweg – de Vries equations and a related many body problem. <...> On a class of polynomials connected with the Korteweg – de Vries equation. <...> Physics 61, 1978, 1–30. 68) On a class of polynomials connected with the Korteweg – de Vries equation. <...> Discrete version of some classical integrable systems and factorization of matrix polynomials. <...> Analogue of the Toda Lattice for Finitely Many Mass Points We consider the analogue of the Toda lattice [8] where only a finite number of mass points are admitted which move freely on the real axis. <...> Thus we can write our system (1.2) as xk = exk−1−xk if we set ex0−x1 condition =0 and exn−xn+1 −exk−xk+1 ,k =1, . , n. x0 = −∞,xn+1 =+∞. (1.2) =0, that is we have the formal boundary (1.3) It is the aim to study <...>
Integrable_Hamiltonian_Systems_and_Spectral_Theory.pdf
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Regular and Chaotic Dynamics, Moscow-Izhevsk
Universitetskaya, 1, Izhevsk, Russia, 426034
Phone: (7–3412) 50–02–95
Fax: (7–3412) 50–02–95
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to reprint the papers included in this volume.
Moser, J¨
urgen
INTEGRABLE HAMILTONIAN SYSTEMS AND SPECTRAL THEORY
c
2003 by Institute of Computer Science, Moscow-Izhevsk
2003 by Regular and Chaotic Dynamics, Moscow-Izhevsk
c
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ISBN 5-93972-274-1
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Printed in the Russian Federation
Стр.2
Contents
Curriculum Vitae . ... .. .. ... .. ... .. .. ... .. ... 4
Editorial Note . . . ... .. .. ... .. ... .. .. ... .. ... 14
Finitely Many Mass Points on the Line under the Influence of an Exponential
Potential — an Integrable System .. .. .. ... .. ... 15
§ 1. Analogue of the Toda Lattice for Finitely Many Mass Points . . . 15
§ 2. Flaschka’s Form of the Differential Equation and Asymptotic
Behavior .... .... ... .... .... .... ... .... . 17
§ 3. Partial Fractions and Continued Fractions . .... ... .... . 21
§ 4. Solution of the Scattering Problem . .... .... ... .... . 27
§ 5. Associated Differential Equations . .... .... ... .... . 34
Three Integrable Hamiltonian Systems Connected with Isospectral Deformations
. . . ... .. .. ... .. ... .. .. ... .. ... 41
§ 1. Introduction . . .... ... .... .... .... ... .... . 41
§ 2. Isospectral Deformations .. .... .... .... ... .... . 45
§3. The n-Particle System on the Line with the Inverse Square Potential 47
§ 4. Asymptotic Behavior, Marchioro’s Conjecture . . ... .... . 50
§ 5. The Periodic Case — Sutherland’s Equation .... ... .... . 53
§ 6. Rational Character of the Solution of (2.4) .... ... .... . 56
§ 7. The Scattering Problem Associated with the Equation of Kac and
Van Moerbeke . .... ... .... .... .... ... .... . 60
Various Aspects of Integrable Hamiltonian Systems .. ... .. ... 65
§ 1. Integrable Hamiltonian Systems . . .... .... ... .... . 65
§ 2. Examples of Integrable Systems, Isospectral Deformations . . . . 68
§ 3. Reduction of a Hamiltonian System with Symmetries . . .... . 71
§ 4. The Inverse Square Potential .... .... .... ... .... . 80
§ 5. Extension of the Geodesic Flow . . .... .... ... .... . 89
§ 6. Geodesics on an Ellipsoid . .... .... .... ... .... . 96
§ 7. An Integrable System on the Sphere .... .... ... .... . 102
§ 8. Hill’s Equation .... ... .... .... .... ... .... . 110
Стр.3
4
Contents
Geometry of Quadrics and Spectral Theory .. .. ... .. .. ... 123
§ 1. Introduction . . .... .... .... ... .... .... .... 123
a. Background . . .... .... ... .... .... .... 123
b. Geodesics on an Ellipsoid . . . . . .... .... .... 124
c. Perturbations of Rank 2 .... ... .... .... .... 127
d. Hyperelliptic Curve . . .... ... .... .... .... 129
e. Applications .. .... .... ... .... .... .... 129
f. Connection with M.Reid’s Result [15] ... .... .... 130
g. Final Remarks . .... .... ... .... .... .... 130
§ 2. Perturbation of Rank 2 .... .... ... .... .... .... 131
a.
b.
Isospectral Manifolds . .... ... .... .... .... 131
Isospectral Deformations ... ... .... .... .... 133
c. The Action of Gl(2,R) ... ... .... .... .... 137
d. Trace Formulae .... .... ... .... .... .... 138
§ 3. Connection with Confocal Quadrics . ... .... .... .... 140
a.
b.
c.
d.
Integrals for the Geodesic Flow on the Ellipsoid . .... 140
Isospectral Deformation ... ... .... .... .... 143
Interpretation of the Eigenvalues and the Frame of L ... 145
Joachimsthal’s Integral .... ... .... .... .... 147
§ 4. The Hyperelliptic Curve . . . .... ... .... .... .... 148
a. The Isospectral ManifoldM(λ) .. .... .... .... 148
b. An Inverse Spectral Problem . ... .... .... .... 152
c. The Symplectic Structure ... ... .... .... .... 156
d. Degenerate Case ... .... ... .... .... .... 159
e. Limit Cases .. .... .... ... .... .... .... 160
§ 5. Examples of Integrable Flows .... ... .... .... .... 162
a. Constrained Systems . .... ... .... .... .... 162
c. A Mass Point on the Ellipsoid Q0(x)+1 = 0 under the
Influence of the Force −ax (Jacobi [6]) .. .... .... 165
b. A Mass Point on the Sphere Sn−1 : |x| =1 under the
Influence of the Force −Ax (C.Neumann [14]) . . .... 164
d. Geodesic Flow on the Orthogonal Group (Manakov [8],
Mischenko [11]) ... .... ... .... .... .... 167
e.
Hill’s Equation (McKean and Trubowitz [9, 10]) . .... 168
§ 6. Appendix ... .... .... .... ... .... .... .... 171
Стр.4
Contents
5
Integrable Hamiltonian Systems and Spectral Theory . ... .. ... 175
§ 1. Introduction . . .... ... .... .... .... ... .... . 175
§ 2. Classical Integrable Hamiltonian Systems and Isospectral Deformations
.... .... ... .... .... .... ... .... . 179
1. Hamiltonian systems .... .... .... ... .... . 179
2.
Integrals .... ... .... .... .... ... .... . 180
3. Perturbation of integrable systems . .... ... .... . 183
4. The inverse square potential .... .... ... .... . 184
5. Constrained Hamiltonian systems . .... ... .... . 186
§ 3. Geodesics on an Ellipsoid and the Mechanical System of
C.Neumann . . .... ... .... .... .... ... .... . 189
1. Geodesic flow on the ellipsoid . . . .... ... .... . 189
2. Confocal quadrics, construction of integrals ... .... . 191
3.
Isospectral deformations .. .... .... ... .... . 193
§ 4. The Schr¨
4. The mechanical problem of C.Neumann . . ... .... . 194
5. The connection between the two systems via the Gauss
mapping .... ... .... .... .... ... .... . 195
6. The Riemann surface .... .... .... ... .... . 199
odinger Equation for Almost Periodic Potentials . . . . 202
1. The spectral problem .... .... .... ... .... . 202
2. The periodic case . . .... .... .... ... .... . 203
3. Almost periodic potential . . .... .... ... .... . 206
4. The rotation number .... .... .... ... .... . 207
5. The Green’s function and a trace formula . ... .... . 208
6. Connection with the KdV equation . .... ... .... . 211
§ 5. Finite Band Potentials ... .... .... .... ... .... . 214
1. Formulation of the problem .... .... ... .... . 214
2. Representation of G(x, x; λ) in terms of partial fractions 215
3. Connection with the mechanical problem . ... .... . 217
4. Solution of the inverse problem .. .... ... .... . 219
5. Finite gap potentials as almost periodic functions .... . 221
6. The elliptic coordinates on the sphere . . . . . . .... . 223
7. Alternative choice of the branch points . . ... .... . 224
§ 6. Limit Cases, Bargmann Potentials . .... .... ... .... . 225
1. Schwarz –Christoffel mapping ... .... ... .... . 225
2. Basis for the frequency module . . .... ... .... . 226
3.
Stationary solutions and their stability behavior . .... . 228
4. The flow on the unstable manifoldW+(en) ... .... . 229
Стр.5
6
Contents
5. The Bargmann potentials ... ... .... .... .... 231
6. A focussing property on S2 . ... .... .... .... 234
7. N-solitons ... .... .... ... .... .... .... 236
8. Concluding remarks . . .... ... .... .... .... 237
DiscreteVersions of Some Classical Integrable Systems and Factorization
of Matrix Polynomials .. ... .. .. ... .. ... .. .. ... 241
§ 0. Introduction . . .... .... .... ... .... .... .... 241
§ 1. The Discrete Version of the Dynamics of a Rigid Body .. .... 246
1.1. The Equations of «Motion» . ... .... .... .... 246
1.2. The Solution of the Matrix Eq. (6): ωTJ −Jω =M . . 249
1.3. Isospectral Deformations ... ... .... .... .... 252
1.4. The Symplectic Geometry of Eq. (6) ... .... .... 254
1.5. The Integration of the Discrete Euler Equation . . .... 258
1.6. Explicit Formulas for the Discrete Dynamics of the 3-Dimensional
Rigid Body .... ... .... .... .... 261
§ 2. The Discrete Dynamics on Stiefel Manifolds and the Heisenberg
Chain with Classical Spins . . .... ... .... .... .... 264
2.1. The Equation of the Dynamics and IsospectralDeformations265
2.2. Discrete Version of the Neumann System and the Heisenherg
Chain With Classical Spins . . .... .... .... 266
§ 3. The Billiard Inside an Ellipsoid . . . . . . .... .... .... 269
3.1. The Splittings and Isospectral Deformations .... .... 270
3.2. Connection Between the Ellipsoidal Billiard and the Discrete
Neumann System .... ... .... .... .... 272
Стр.6