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§1. Introduction 181
be expressed in terms of Abelian integrals, as Jacobi discovered in 1838 and
they can indeed be related to the finite gap potentials. This observation is due
to Trubowitz and the writer and has been presented in brief form in Moser 1980
[22], Al’ber 1981 [3], Veselov 1980 [29]. Here we want to give a selfcontained
derivation and describe related questions like the Bargmann potentials which
are obtained from the finite gap potentials as limit cases when the intervals I
I
, ..., I
1
shrink to g different points.
g−1
0
Actually the finite gap potentials will be related to a different mechanical
2
problem, namely, the motion of a mass point on an n-dimensional sphere x
+ x
2
+ ··· + x
1
2
=1under the influence of a force created by a quadratic
n
+
0
potential. Also this problem can be solved in terms of Abelian integrals, as
was shown by C. Neumann in 1859. He used the same technique of separation
of variables in the Hamilton – Jacobi equations which had been developed by
Jacobi and used by him for finding the geodesics on the ellipsoid. However,
only recently it was found by H. Kn¨orrer [13], that Neumann’s problem can be
reduced to Jacobi’s geodesic problem by using the Gauss map, i. e. by mapping
the ellipsoid via the exterior unit normals on the unit sphere. This will be
described in Section 3.
We wish to indicate briefly how the connection between the spectral and the
mechanical problem is brought about. This relation is indeed rather unexpected
and is completely different from the familiar quasiclassical limit which also
relates a quantum mechanical problem to a classical one. We recall that the set
of potentials belonging to a finite band spectrum I
g
torus T
. Moreover, if q = q(x) belongs to Tgso do its translates q(x + t) for
, I2, ..., Igis a g-dimensional
0
any real t. This defines a flow on the torus taking q(x) into q(x + t),which
defines the function q(t) in terms of q(0). It turns out the torus T
g
together
with this flow can be mapped onto an integral surface of the mechanical problem
together with the flow of the Neumann problem. In this way the inverse problem
is completely reduced to the mechanical problem and formulae for the finite band
potentials can be obtained. In general, they are given by almost periodic functions
representable as power series in exp(±iω
x), exp(±iω2x), ..., exp(±iωgx).
1
This procedure lends itself to other, non-almost periodic potentials suggested from the mechanical problem: This problem admits equilibrium solutions
possessing stable and unstable manifolds. The orbits on these invariant manifolds
clearly approach the equilibrium solutions at an exponential rate, and therefore
are not almost periodic. They give rise to potentials decaying at exponential
rate at infinity. They turn out to be given by rational functions of real exponentials exp(κ
,x), exp(κ2,x), ..., exp(κgx) and are nothing but the Bargmann
1
,
182 Integrable Hamiltonian Systems and Spectral Theory
potentials
q(x)=−2
with
Δ=detδ
ij
−
κi+ κ
2
d
log Δ
dx
η
iηj
j
i, j=1, ...,g
ηj= ajexp(−κix); 0 < κ1< κ2<... <κg.
This form for fixed κ
with continuous spectrum from 0 to +∞ and g point eigenvalues at λ = −κ
, κ2, ..., κgis a g-dimensional manifold of potentials
1
j
They can be viewed as limit cases of the almost periodic potentials when the
intervals I
− 1(j =1, 2, ..., g) shrink to the points −κ
j
2
.
j
But there are other limit cases where only some intervals collapse to points
and which correspond to the stable and unstable manifolds of periodic orbits or
lower dimensional invariant tori of the mechanical problem. The corresponding
potentials are asymptotically periodic or almost periodic as x →±∞. Thus
this connection between spectral theory and this mechanical problem provides a
number of new explicit eigenvalue problems with known spectrum.
If we ignore these limit cases the finite band potentials for a given spectrum
I
, I1, ..., Igare almost periodic and form a torus Tg. Besides the translation
0
q(x) → q(x + t) there are g −1 other flows defined on T
g
leaving the spectrum
fixed.
Just like the translation in generated by the differential equation
∂q
∂q
=
∂t
∂x
2
.
one of the other flows is generated by the nonlinear equation
∂q
∂t
− 6q
∂q
∂x
3
∂
q
+
=0 (1.2)
3
∂x
which is the well known Korteweg– de Vries equation. It has the distinguished
2
property that for any solution q = q(x, t) the spectrum of −(d
/dx2)+q(x, t)
is independent of t. Therefore one speaks of isospectral deformations. In
fact, the study of the Korteweg – de Vries equation, which was initiated by
M. Kruskal and his coworkers [10], was largely based on the construction of
conserved quantities by interpreting the flow generated by the KdV equation as
an isospectral deformation of the Schr¨odinger equation.
§2. Classical Integrable Hamiltonian Systems and Isospectral Deformations 183
It turns out that also the classical problems of Jacobi and Neumann can
be approached and solved by use of isospectral deformations of some classes
of matrices instead of the separation of variables in the Hamiltonian – Jacobi
equation.
In the next two Sections we will give the definition and properties of
Hamiltonian systems in the framework of classical mechanics and describe the
geodesic flow on the ellipsoid and Neumann’s problem from this point of view. In
Section 4 we discuss the Schr¨odinger operator for almost periodic potentials. In
this case the familiar Floquet theory of periodic systems of differential equations
fails, but we will introduce a substitute for the Floquet multiplier μ = μ(λ) for
the almost periodic eigenvalue problem
+ qϕ = λϕ.
−ϕ
This quantity is related to the so-called density of states α = α(λ), a monotone
increasing continuous function on the real axis, which determines the spectrum
of the operator: The support of the measure dα agrees with the spectrum.
On the other hand the Floquet multiplier can also be used to determine the
conservation laws of the KdV equation as we will elaborate in Section 4.
Finally in Section 5 we discuss the finite band potentials and their relation
to the classical problems of Jacobi and Neumann. In Section 6 we study limit
cases, the relation of the Floquet multiplier to the Christoffel –Schwarz map of
the upper half plane to a slit domain as well as the Bargmann potentials.
§ 2. Classical Integrable Hamiltonian Systems and Isospectral
Deformations
1. Hamiltonian systems
In classical mechanics one describes the equation of motion by Hamiltonian
systems of the form
dq
j
∂H
=
dt
∂p
where q =(q
, ..., qn) ∈ Rn, p =(p1, ..., pn) ∈ Rnand H = H(q, p) is a
1
smooth function on an open domain Ω of R
dp
j
∂H
dt
x =
= −
q
p
, (j =1, ..., n)(2.1)
∂q
j
2n
. Combining q, p to a single vector
2n
∈ R
,
j
184 Integrable Hamiltonian Systems and Spectral Theory
we write this system also in the form
dx
= JH
dt
x
(2.2)
where H
x
=(H
, ..., H
x
1
) is the gradient of H and
x
2n
J =
0 I
−Ix0
x
.
It is customary to introduce the «Poisson bracket» {F, G} for two functions
1
F , G ∈ C
(Ω) by
{F, G} =
n
j=1
∂F
∂q
∂p
j
∂G
∂F
∂G
−
∂p
j
)(2.3)
∂q
j
j
so that the equations of motion (2.1) or (2.2) can be written in the form
dx
j
= {x
,H} (j =1, 2, ..., 2n).
dt
j
The system of differential equations (2.1) is defined in terms of a single
function H ∈ C
n
(Ω). Using a notation borrowed from differential geometry
we associate with (2.1) also the «vectorfield» or first order partial differential
operator
X
These Hamiltonian vectorfields with H ∈ C
commutator [X
]=XFXG− XGXFis again a differential operator of
F,XG
n
=
H
j=1
H
∂
− H
p
j
∂q
j
∞
(Ω) form a Lie algebra since the
∂
q
.
j
∂p
j
first order generated by a Hamiltonian. As a matter of fact,
[X
F,XG
]=−X
. (2.4)
{F, G}
It is this Lie algebra of Hamilton vectorfields which is the object of classical
mechanics. It can be generalized to vectorfields on symplectic manifolds but this
extension will not be needed in these lectures.
2. Integrals
The important concept of an integral of a Hamiltonian vectorfield X
H
defined as follows:
is
§2. Classical Integrable Systems and Isospectral Deformations 185
1
Definition 1. A non-constant function F ∈ C
F = {F, H} =0.
X
H
(Ω) is called an integral if
This implies clearly, that F(x(t)) is independent of t,i.e. F(x) is constant
along orbits. Since {·, ·} is an alternating form, i. e. {F, G} = −{G, F } it
follows that {H, H} =0,i.e. H is always an integral. Moreover, one has the
symmetric statement: If F is an integral for X
then H is an integral for XF.
H
The existence of one or more integrals can be used evidently to reduce the
system to a simpler one and therefore the knowledge of integrals is of interest.
For example, if one has 2n − 1 integrals F
independent then the equations F
= cjdefine the orbits of the system. Actually
j
for which the gradients are linearly
j
for Hamiltonian systems it suffices to know n «commuting» integrals to render
the system integrable.
Definition 2. A Hamiltonian vectorfield X
grable» if it possesses n integrals F
,H} =0
i) {F
j
ii) {F
iii) The gradients dF
j,Fk
} =0
are linearly independent in Ω.
j
j
The first condition expresses that the F
∈ C1(Ω) satisfying
in Ω ⊂ R2nis called «inte-
H
are integrals, the second is ex-
j
pressed by saying that they commute since by (2.4)
,X
[X
F
]=0
F
j
k
and the third condition requires a nondegeneracy which we will have to relax
frequently.
Here are two simple examples of integrable systems:
XAMPLE A). The linear system describing n oscillators
E
n
H =
1
2
j=1
(p
2
j
+ ω
2
2
q
); ωj> 0
j
j
clearly possess the integrals
2
2
+ ω
2
q
j
j
= p
F
j
j
satisfying i) and ii). The condition iii) is violated on
= {(p, q) ∈ Rn: pk= qk=0}
S
k
186 Integrable Hamiltonian Systems and Spectral Theory
but is satisfied on
2n
Ω=R
E
XAMPLE B). The second example is given by
H = H(p
where F
= pjare n integrals satisfying i)–iii).
j
If in the physical interpretation the variable q
gles (say, mod2π) one calls the q
k
n
,
, ..., pn)
1
-
k=1
.
S
k
have the meaning of an-
k
«angle variables» and the pkthe «action
variables».
For an integrable Hamiltonian system the vectorfield X
gential to the manifolds M
= {(q, p): F1= c1,F2= c2, ..., Fn= cn},i.e.
c
these manifolds are invariant under the flow generated by X
is clearly tan-
H
. Thus the phase
H
space is foliated into these n-dimensional invariant manifolds. According to
a theorem by V. I. Arnold, and a sharpened version by R. Jost, such a leaf is
n
necessarily a torus T
provided it is compact and connected. Moreover, in the
neighborhood of such a compact leaf one can introduce action angle variables,
say P , Q, via a canonical transformation, such that Q
variables, the P
, ..., Pnare the action variables and H = H(P1, ..., Pn).
1
, ..., Qnare the angular
1
In other words, in a neighborhood of such a compact leaf an integrable
system always has the form of example b). In the example a) the leaves are
given by
which for c
> 0, ..., cn> 0 are indeed tori Tn. Moreover, on such an
1
p
2
+ ωjq
j
2
= cj(j = i, ..., n)
j
invariant compact leaf the differential equation becomes linear. Indeed, in the
action-angle variables P , Q the differential equations become
˙
Q
= H
j
,˙Pj= −H
P
j
=0
Q
j
which are solved by
Q
(t)=H
j
(P )t + Qj(0),Pj(t)=Pj(0).
P
j
Thus the integration of such integrable systems (with compact leaves) becomes
trivial, which explains the name.
Actually the commuting integrals F
F
= ϕk(F1, ..., Fn)
k
can be replaced by other integrals
j
§2. Classical Integrable Systems and Isospectral Deformations 187
which are still commuting since
{F
,Fl} =
k
∂(ϕk,ϕl)
∂(Fi,Fj)
i, j
{F
i,Fj
} =0.
Therefore integrals themselves are not of primary interest but the foliation defined
by the Pfaffian system
dF
=0,dF2=0, ..., dFn=0
1
is more important. It defines the foliation which has a geometrical meaning. In
the following we will frequently make use of this freedom of replacing a set of
integrals by functions of these.
3. Perturbation of integrable systems
It is important to realize that among the Hamiltonian systems the integrable
ones are exceptional. If an integrable system is perturbed slightly the integrals
are in general destroyed. This is related to the often quoted «theorem» of
Poincar´e’s about the nonexistence of integrals. In 1923 E. Fermi [8] extended
Poincar´e argument to show that on a fixed energy surface there would be no
other integrals after general perturbations. However, his argument is only formal
and uses the erroneous assumption that for a smooth Hamiltonian system the
invariant sets are smooth also. Even though the integrals may get lost after
perturbation, the whole foliation does not get lost, but a large subset of the
tori (namely the non-resonant tori) do survive small perturbations and form a
complicated invariant Cantor set of positive measure. This is the content of the
so-called K. A. M. theory which establishes the existence of this set of invariant
tori. In a recent paper J. P¨oschel [26] showed that actually the perturbed system
can still be viewed as an integrable one when restricted to this Cantor set. This
means that one can define n integrals in involution on this Cantor set which are
differentiable in the sense of Whitney.
However, we will not study such perturbed systems and consider Hamiltonian systems which are integrable in an open set Ω of R
2n
. Even if a system is
integrable it is frequently not easy to find a set of integrals. As a typical example
we will discuss in the next section the geodesic flow on an ellipsoid.
Here we mention an example which is equivalent to the Kepler problem
n
, and is given by the Hamiltonian
in R
H =
1
2
|p|
2
−|q|−1.
188 Integrable Hamiltonian Systems and Spectral Theory
This system is clearly rotation symmetrical and therefore has many integrals,
namely p
− pjqifor 1 i<j n. But these are not commuting integrals,
iqj
and it takes some experimenting to find the commuting integrals
F
=
k
1<i<j<k
(piqj− pjqi)2(k =2, 3, ..., n)
F
= H;
1
thus the Kepler problem in R
the functions H
, H2, F3are related to the Delauney variables of Celestial
1
n
is indeed integrable. Incidentally, for n =3
Mechanics.
4. The inverse square potential
As an example with non obvious integrals we consider the following exam-
1
ple due to Calogero (1971)
. Consider n masspoints of equal mass, say =1,on
the line repelling each other with a force proportional to the third power of the
reciprocal of the distance. If q
is the position of the j-th masspoint the motion
j
is given by
2
q
d
j
∂U
where
U =
1<i<j<n
= −
dt
2
∂q
j
V (qi− qj)V (x)=
|x|
b
.
2
(2.5)
The Hamiltonian is, of course, given by
1
H =
2
+ U (q).
p
j
2
For this problem n rational integrals were discovered by M. Henon, Flaschka
and Manakov [11, 9, 16]. However, in this case the integrals are found in a
surprising way, namely as the eigenvalues of the matrix
L(q, p) = diag(p
1
For reference, see [21], [22], [23].
, ..., pn)+i
1,p2
1 − δ
qj− q
jk
ki, k=1, ..., n
. (2.6)
§2. Classical Integrable Systems and Isospectral Deformations 189
One shows that for any solution of the system (2.5) with p
=˙qjthe matrix
j
L(t)=L(q(t), (t)) is similar to L(0) i. e., there exists a unitary matrix U = U(t)
with
−1
U(t)
L(t)U(t)=L(0). (2.7)
This clearly implies that the eigenvalues of L(t) are integrals. The same is true
for the symmetric functions of the eigenvalues given by
F
(q, p)=tr(Lk)(k =1, 2, ..., n)(2.8)
k
which are clearly rational functions in q
that the F
commute (see Moser [21]).
k
, pj=˙qj. Moreover, it can be shown
j
In order to verify the validity of (2.7) we define U (t) by a differential
equation
dU(t)
= B(t)U(t); U (0) = I (2.9)
dt
where B = B(t) is skew-Hermitian. By differentiation of (2.7) we arrive at
d
−1
U(t)
L − BL + LBU(t)=0
dt
or
d
L =[B, L]. (2.10)
dt
If we can find B = −B
∗
in such a way that (2.10) holds as a consequence
of (2.6) then (2.9) and (2.10) implies (2.7) by integration.
Now one verifies by calculation that (2.10) holds if we set
B = i diag(d
1,d2
d
k
=
, ..., dn) − i
n
j=1
j=k
1
(qj− qk)
1 − δ
(qj− qk)
.
2
jk
2
Thus the differential equations (2.6) give rise to an isospectral deformation
for the matrix L = L(q, p) given by (2.6). One has to keep in mind that L, B
were guessed! A more systematic derivation of these formulae is duo to Kazhdan,
Kostant and Sternberg. For an exposition and a discussion of the solutions, in
particular their scattering behavior, see [20, 22]. We mention that in this case
190 Integrable Hamiltonian Systems and Spectral Theory
the leaves of the foliation Fk= ckare not compact but equivalent to Rn.We
described this example to illustrate the idea of isospectral deformations which we
will use again in the next section. For the Calogero system no other technique
has been found to construct the integrals. For n =3the problem was considered
by Jacobi by separation of variables but this approach fails for n>3.
The idea to relate the differential equation to isospectral deformations was
developed by P. D. Lax [14, 15] in connection with the Korteweg – de Vries
equation. After previous work by Kruskal et al., he noticed that the KdV (1.2)
can be written in the form
d
L =[B, L]
dt
where
L = −D
B =4D
2
+ q(x, t),D=
3
− 3(qD + Dq).
d
dx
Thus the eigenvalues of the Shr¨odinger operator L under appropriate boundary
conditions give rise to integrals for the Korteweg – de Vries equation! We will
come back to this equation in Section 5 where we derive the integrals of the KdV
equation in the framework of almost periodic functions from the generalized
Floquet multiplier μ(λ).
5. Constrained Hamiltonian systems
In the following we will frequently consider Hamiltonian systems on submanifolds M of R
The dimension of M is 2n −2r it the dG
2n
,givenby
G
(x)=...= G2r(x)=0. (2.11)
1
are linearly independent on M.We
j
will require more by assuming
det({G
j,Gk})j, k=1, ..., 2r
=0, (2.12)
which makes M a symplectic manifold.
It a given system (2.2) defines a vectorfield tangential to M then there is
no difficulty in restricting this system to M . The conditions for this to be so is
clearly
= −{H, Gj} =0 for j =1, 2, ..., 2r
X
HGj
on M.
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