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§6. Limit Cases, Bargmann Potentials 241
or
δ
x − 4κ
2
t = −
2
2
κ
2
=const.
These values of x correspond to the focus where we have by (6.21)
2
κ
2
ψ
0
1
=
,ψ1=0,ψ
κ
2
2
=1−
2
κ
1
κ
2
2
and therefore
q = −2κ
ψ
− 2κ
1
1
ψ
2
2
= −2(κ
2
2
2
− κ
2
)
1
2
2
2
proving our claim.
It would be interesting to find a generalization of this result to the N -soliton
N
and an analogue focussing for the mechanical problem on S
.
8. Concluding remarks
With these considerations we wanted to show how the interplay between
the inverse spectral problem and the mechanical problem on the sphere can be
used to advantage. We found the explicit orbits on the stable manifolds starting
from the formulae for the Bargmann potentials. These potentials appeared as
the limit case of the finite band potentials where all bands collapse to points,
the eigenvalues of this potential. But this interplay can be used to study further
interesting potentials for which the spectrum can be determined explicitely. For
example, in the neighborhood of one of the stationary solutions e
with purely
k
imaginary and real eigenvalues one finds unstable periodic orbits. These periodic
orbits in turn have stable and unstable manifolds and they can be described in
terms of exponents exp(r
x), rjreal, and purely imaginary exponentials, due to
j
the periodic behavior. Therefore the corresponding potential will exponentially
approach periodic orbits as x →±∞. Their spectrum will be given by two
bands (one finite one semi-infinite) and point eigenvalues corresponding to the
real exponentials. Similarly, one can consider unstable invariant tori, and the
corresponding stable and unstable manifolds (whiskers) corresponding to potentials which are asymptotic to quasi-periodic functions as x →±∞.Theseare
simply limit cases of the quasi-periodic potentials discussed in the proceeding
Section.
Added in proof : Since those lectures were given the interesting papers
B. M. Levitan, Almost periodicity of infinite-zone potentials, Math. USSR, 18 (1982),
249–273;
242 Integrable Hamiltonian Systems and Spectral Theory
B. M. Levitan, Approximation of infinite-zone potentials by finite-zone potentials.
Izv. Akad. Nauk USSR, ser. math., 46, №1 (1982). 56–87;
appeared. They extend the construction of the finite-band potential to the case of infinitely
many bands. In the periodic case this had been done by McKean – Trubowitz [18] but
Levitan’s approach is in contrast to [18] based on the study of the Jacobi inversion in the
infinite genus case. These papers contain further interesting related references.
References
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Lie Algebras and Curves, Advances in Math., 38 (1980), 267–317.
[2] M. Adler, and P. van Moerbeke, Linearization of Hamiltonian Systems, Ja-
cobi Varieties and Representation Theory, Advances in Math., 38 (1980),
318–379.
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Type, Comm. Pure Appl. Math., 34 (1981), 259–272.
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Appl. Math., 32 (1979), 121–125.
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Korteweg – de Vries type, finite zone linear operators and Abelian varieties,
Russian Math. Surveys, 31 (1976), 59–146.
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Mat. Nauk, 14 (1959) 57–119.
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quasiergodisch ist, Phys. Z., 24 (1923), 261–265.
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[10] C. H Gardner, J. M. Greene, M. D. Kruskal, R. Miura, Korteweg – de Vries
equation and generalization VI. Methods for exact solution, Comm. Pure
Appl. Math., 27 (1974), 97–133.
[11] M. Henon, Integrals of the Toda lattice,Phys.Rev.B,9 (1974), 1921–1923.
References 243
[12] R. Johnson, and J. Moser, The rotation number for almost periodic poten-
tials, Comm. Math. Phys., 84 (1982), 403–438.
[13] H. Kn¨orrer, Geodesics on Quadrics and a Mechanical Problem of
C. Neumann, J. Reine Angew. Math., 334 (1982), 69–78.
[14] P. D. Lax, Nonlinear partial differential equations of evolution, Proc. Int.
Cogr. Math. (Nice 1970), Gauthier-Villars, 831–840, Paris 1970.
[15] P. D. Lax, Periodic solutions of the KdV equation, Comm. Pure Appl. Math.,
28 (1975), 141–188.
[16] S. V. Manakov, Complete integrability and stochasticity for discrete dynam-
ical systems, Journal E. Theor. Phys., 40 (1974), 269–274.
[17] H. P. Mc Kean, and P. van Moerbeke, The spectrum of Hill’s equation,In-
vent. Math., 30 (1975), 217–274.
[18] H. P. Mc Kean, and E. Trubowitz, Hill’s operator and Hyperelliptic function
theory in the presence of infinitely many branch points, Comm. Pure Appl.
Math., 29 (1976), 143–226.
[19] H. P. Mc Kean, Integrable Systems and Algebraic Curves, Lecture Notes in
Math., 775, 83–200, Springer-Verlag, 1979.
[20] J. Moser, The scattering problem for some particle systems on the line, Lec-
ture Notes in Math., 597, Geometry and Topology, 441–463, Springer-Verlag, 1977.
[21] J. Moser, Three integrable Hamiltonian systems, Advances in Math., 16
(1975), 197–220.
[22] J. Moser, Varios Aspects of Integrable Hamiltonian Systems, Progress in
Math., 8, 233–289, Birkh¨auser, Boston, 1980.
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sium 1979, Springer-Verlag, 1980.
[24] J. Moser, An example of a Schr ¨odinger operator with almost periodic po-
tential and nowhere dense spectrum, Comment. Math. Helv., 56 (1981),
198–224.
244 Integrable Hamiltonian Systems and Spectral Theory
[25] D. Mumford, An algebro-geometric construction of commuting operators
and of solutions to the Toda lattice equation. Korteweg – de Vries equation
and related nonlinear equations, Proc. Int. Symp. On Algebraic Geometry,
Kyoto 1977, 115–153, Tokyo 160–91, Japan.
[26] J. P¨oschel, Integrability of Hamiltonian Systems on Cantor Sets, Comm.
Pure Appl. Math., 35 (1982), 653–696.
[27] G. Scharf, Fastperiodische Potentiale, Helv. Phys. Acta. 24 (1965),
573–605.
[28] R. Schrader, High energy behavior for non-relativistic scattering by station-
ary external metrics and Yang – Mills potentials, Z. Physik C, Particles and
Fields, 4 (1980), 27–36.
[29] A. P. Veselov, Finite band Potentials and an integrable Systems on the
sphere with quadric Potentials, Functional Anal. Appl., 14 (1980), 48–50
(Russian).
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die zugeh¨origen Entwicklungen willk ¨urlicher Funktionen, Math. Ann., 68
(1910), 220–269.
Discrete Versions of Some Classical Integrable
), k ∈ Z
k
n
×
1
n
Systems and Factorization of Matrix Polynomials
§ 0. Introduction
In this paper we study a class of discrete integrable systems which are closely
related to problems occurring in mathematical physics such as the Heisenberg
model for classical spins or the billiard problem in the interior of an ellipsoid.
A discrete system can be viewed as the iterates of a symplectic mapping, the
time t ∈ Z being the number of iterations. Such a system will be called integrable
if it possesses sufficiently many integrals which are in involution with respect to
a symplectic structure.
To describe such a discrete system we take as starting point a variational
principle
δS =0
for a functional S = S(X ) defined on the space of sequences X =(X
by a formal sum
Here X
are points on a manifold
k
The Euler – Lagrange equation of such a functional are second order difference
equations (see Sect. 1) and k ∈ Z plays the role of the discrete time.
This description is to be viewed as the discrete analogue of the Hamilton
principle δS =0for
S =
S =
(Xk,X
k∈Z
n
andis a function on Q2n=
k+1
) .
(q, ˙q) dt,
.
and the related symplectic flow can, as usual, be defined via the Legendre trans-
form provided det(
2n
on Q
under an appropriate nondegeneracy assumption (see Sect. 1, Sect. 2,
) =0. Similarly, one can define a sympleclic structure
˙q ˙q
(8) and [1]). We will call such a system «integrable» if there are sufficiently
many integrals which are in involution with respect to this symplectic structure.
1
Jointly with A. Veselov. Discrete version of some classical integrable systems and factorization
of matrix polynomials. Commun. Math. Phys. 139, 1991, 217–243.
246 Discrete Versions of Some Classical Integrable Systems
As an example of this setup we mention the Heisenberg chain with classical
spins, where
2
M = S
= {x ∈ R3, |x| =1}
and
=(x, J y),
where J is a symmetric matrix, which we may take to be diagonal.
The corresponding chain of quantum spins
1
(so-called XY Z Heisenberg
2
model) was investigated by Faddeev and Takhtajan [2] in the framework of the
quantum inverse scattering method, using the fundamental results by Baxter [3].
As it was shown by Pokrovsky and Khokhlachev [4], the problem of finding
some special eigenfunctions in the quantum XYZ-model leads to the stationary
equation δS =0for the Heisenberg chain with classical spins. For this discrete
system Granovsky and Zhedanov [5, 6] found two algebraic integrals and special
solutions. The integrability of these systems, even for arbitrary dimension n
n
of the sphere: M = S
, was shown by one of the authors ([7], see also [1]),
where the general solution was described in terms of θ-functions, generalizing the
connection between the spectral theory of one-dimensionalSchr¨odinger operators
and the classical Neumann systems derived in [8, 9].
The main problem to be discussed in the first part of this paper is a chain
n
of orthogonal matrices: We take
(X, Y )=tr(XJYT),
= O(N), n = N (N −1)/2,and
where J is a positive symmetric matrix
1
. This problem was introduced in [1]
where it was shown that in the continuous limit this problem leads to the Euler
problem for force-free motion of a rigid body as it was generalized by Arnold lo
arbitrary dimensions.
1
Alternately, one could use the positive Lagrangian
T
). Indeed, for X, Y ∈(N) this expression agrees with tr J −tr(XJYT),
−Y )
i. e. differs from −
(X, Y) only by a constant.
tr((X − Y )J(X −
2
For the cases N =3, N =4this system was shown to be integrable by
explicit construction of commuting integrals [1]. In Sect. 3 we will establish
the integrability of this system for all N by using the discrete version of the
isospectral technique, which leads to the complete description of the dynamics
1
We denote the «moment of inertia» by J and not, as is customary, by I, to avoid confusion with
the identity matrix
§0. Introduction 247
of this system in terms of Abelian functions. The flow is quasi-periodic in the
discrete time parameter k and is linear on a Prym variety.
We indicate the approach underlying the solution of this problem. It is
based on the construction of an isospectral mapping on a class of matrices (L)
into itself, analogous to the Lax approach in the continuous case in which the
differential equation is cast in the form˙L =[L, A] for some class of linear
operators (L).
Finding the class of matrices (L) and the isospectral deformation is a
hit-ormiss game and depends on good guesses. In the discrete case it turns
out to be connected with a factorization of matrices. We recall the beautiful
observation by Symes [11] that the QR-algorithm of Jacobi matrices is closely
linked to the Toda flow: The QR-algorithm, an important device in numerical
analysis for diagonalization of matrices, consists in factoring a real nondegenerate matrix L into a product L = QR of an orthogonal matrix Q and an upper
triangle matrix R with positive diagonal elements. Now the mapping
L = QR → L
= RQ = Q1R
1
1
gives rise to an isospectral map since L1= Q−1LQ. It was Symes’ observation
[11] that the application of this process to L =expK,whereK is a symmetric
tridiagonal (or Jacobi-) matrix leads to an integrable mapping which is interpolated by the Toda flow. This idea has been extended to more general classes of
matrices by Deift et al., see [12].
Although this mapping has no variational description, this idea turns out to
be fruitful also for the problems at hand. The main new feature is that we start
with a class of certain quadratic matrix polynomials
L(λ)=A
+ A1λ + A2λ
0
2
and a suitable factorization
L(λ)=(B
+ B1λ)(C0+ C1λ)=B(λ)C(λ).
0
If such a factorization exists and can be defined in a unique way then it gives
rise to an isospectral mapping L(λ) → L
L
(λ)=C(λ)B(λ)=B1(λ)C1(λ)=B−1(λ)L(λ)B(λ).
1
(λ) by exchanging the factors:
1
This can be viewed as a discrete analogue of the Lax representation.
248 Discrete Versions of Some Classical Integrable Systems
The main difficulty is, of course, to find the class of matrix polynomials
together with a factorization in such a way that it corresponds to the dynamics of
the given problem. Moreover, even if one has such a factorization it is generally
not unique and the above procedure gives rise only to a correspondence, i. e. a
multiple valued mapping; this is in good agreement with the fact that frequently
the difference equation δS =0gives rise to such correspondences.
In the problem of orthogonal chains we are able to describe such a class of
matrix polynomials and a corresponding factorization which can be made unique
by specifying a suitable splitting of the spectrum. From this Lax representation
we will find the integrals as well as the algebraic curve on whose Jacobian
variety the flow becomes linear in k. A general theory of factorization of matrix
polynomials can be found in [13]. The special factorization derived here (see
Sect. 1) uses standard ideas from [13]; it involves orthogonal matrices and may
be of interest in itself. Using the ideas of «finite-gap» integration [14] in the
matrix case developed by Dubrovin [15] we exhibit explicit formulas for the
dynamics in the case n =3in terms of classical elliptic functions.
In Sect. 2 we discuss some generalizations of this system, including the
above mentioned Heisenberg model with classical spins. We take
manifold of rectangular n × N matrices X , n N for which XX
T
= Inand
as the
define the Lagrange function by
(X, Y )=tr(XJYT),
where J is a symmetric N ×N matrix. In other words
of orthonormal n-frames in RN.Forn =1, N =3this represents the
V
n, N
is the Stiefel manifold
Heisenberg model and for n = N we obtain the chain of orthogonal matrices
described above. For n =1we show how the factorization procedure leads
to the hyperelliptic curves and formulas involving θ-functions as in [1]. In the
general case 1 <n<N we exhibit the corresponding factorization problem
without full treatment.
The last section (Sect. 3) is devoted to the billiard problem in the interior
of an ellipsoid in R
N
. This problem also fits into the above framework. The
relevant class of matrices for the Lax representation agree with those introduced
in [10] for the study of the geodesic flow on an ellipsoid — probably the oldest
nontrivial integrable system in arbitrary dimensions. This class of matrices fits
into the procedure of Sect. 1. Finally, we will establish a connection between
this billiard problem and a discrete version of the Neumann problem, where a
certain symmetry of this system, found in [1], will play a crucial role. In the
continuous case such a connection between the geodesic flow on the ellipsoid
§0. Introduction 249
and the Neumann system was discovered by Kn¨orrer with the help of the Gauss
mapping of the ellipsoid [16]. However, the connection described here is of a
different nature.
In the above discussion we referred frequently to a discrete version of a
continuous system, as in the case of the billiard problem inside an ellipsoid and
the geodesic flow on the same ellipsoid — whose orbits are obtained as limits
of tangential billiard shots. Another example is the above mentioned chain of
orthogonal matrices and the corresponding continuous system of the force free
top. Without trying to be precise we require in all these cases a) that the discrete
system tends to the continuous system under a limit process and b) that both
systems are integrable and are given by «natural» variational problems.
As a rule it is easy to go from a discrete system to a continuous one without destroying integrability. However, the converse is much more difficult, as
is often the case if one wants to preserve some structure under discretization
of a continuous system. Of course, one could take the «time ε» mapping of
a flow but that is usually not described by a simple variational problem. The
difficulty is to preserve the integrability under discretization. In this sense our
method may be of interest since it provides an approach to the construction
of an integrable discretization for the continuous system with known Lax representation, polynomially depending on an additional «spectral» parameter λ.
The importance of representations of this type became clear after the paper of
Novikov [17]. They exist for most of the known integrable hamiltonian systems
and are related to the theory of Lie algebras (see [18, 19]). From this point of
view the nature of the factorization procedure calls for a better understanding.
Notice that in all our examples the matrix polynomials L(λ) have the property
∗
(λ)=L(λ),whereL∗(λ)=LT(−λ) and the corresponding factorizations
L
L(λ)=B(λ)C(λ) satisfy the condition B(λ)=C
∗
(λ).
In a forthcoming paper [29] it is shown that the factorization of certain
linear(!) matrix polynomials, introduced in [10], leads to the billiard problems in
domains on the sphere and the Lobachevsky space, bounded by conic sections.
The present paper was completed in February 1989 and has been circulated
as a preprint of the Forschungsinstitut f¨ur Mathematik Z¨urich. For various
reasons its publication has been delayed. We added some relevant new references
at the end of this revised paper. In particular, we draw attention to the interesting
work by Deift, Li, and Tomei [32] in which the systems considered in the
present paper are related to loop groups. Moreover, it is shown how the discrete
mappings considered here can be interpolated by integrable Hamiltonian flows.
250 Discrete Versions of Some Classical Integrable Systems
§ 1. The Discrete Version of the Dynamics of a Rigid Body
1.1. The Equations of «Motion»
We consider the functional S(X ), determined by a formal sum
S =
tr(XkJX
k
T
)(1)
k+1
on the sequences X =(X
) with Xk∈ O(N ), i. e. orthogonal N by N matrices.
k
The stationary points of S are described by the equation δS =0or
X
J + X
k+1
where Λ
a way that X
k
T
=Λ
is a matrix Lagrange multiplier, which is determined in such
k
T
X
= I. Λkis uniquely determined by X
k
k
as we will see later, not uniquely by X
of X
k−1
, Xk.
J =ΛkXk, (2)
k−1
, Xk, X
k−1
, Xk; it is a complicated function
k−1
k+1
but
Therefore we use the Euler description of the dynamics. This can be done
in the following way. Rewrite (2) as
X
k+1
Introducing m
that m
= mk. The conservation of mk, which is the discrete analogue of
k+1
T
JX
k
= XkJX
k
+ X
k−1
JX
T
k−1
T
=Λk=Λ
k
− X
k−1
T
= XkJX
k
T
JX
we see that the last Eq. (3) means
k
T
k+1
+ XkJX
T
k−1
. (3)
the angular momentum in space [20] is the consequence of the left-invariance
(X, Y ) (see [1]). In the variables fixed relative to the body we have the
of
«angular velocity» ω
k
= X
T
X
k−1
k
tum with respect to the body» M
= X
k
= X
−1
k
X
−1
k−1
∈ O(N ) and «angular momen-
k−1
mkX
k−1
T
= ω
J − Jωk∈ O(N )
k
and thus Eq. (3) can be rewritten as a «discrete Euler –Arnold equation» [1]
M
= ωkMkω
k+1
= ω
M
k
In the continuous limit when X
≈ I −εΩ(t
= εM(t
), ωk= X
k
), M = JΩ+ΩJ , this Eq. (4) becomes the usual Euler –Arnold
k
k
−1
X
k−1
T
k
k
−1
,
k
J −Jωk,ωk∈ O(N ).
= X(tk), tk= t0+ kε, ωk= X
and Mk= ω
T
J − Jωk≈ ε(JΩ+ΩJ)=
k
−1
k
X
k−1
(4)
≈
equations for the motion of the N -dimensional rigid body
˙
M =[M, Ω],
M = JΩ+ΩJ, Ω ∈ o(N).
(5)
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