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§7. An Integrable System on the Sphere 111
where we dropped the unessential constant
1
. Thus the equations of
a
ν
2
motion become
˙u
= H
j
, ˙vj= −H
v
j
u
j
and the Hamilton – Jacobi equations
∂S
∂u
=const.
Hu,
More precisely, we ask for a solution S = S(u, η) depending on η =
, ..., ηn) of the equation
=(η
1
∂S
∂u
= η
. (13)
1
Hu,
Then the canonical mapping (u, v) → (ξ, η) defined by
v
=
j
takes the Hamiltonian into H = η
˙
ξ
j
∂S
,ξj=
∂u
j
and the differential equation into
1
= δ
, ˙ηj=0. (15)
j
1
∂S
∂η
j
This is the standard use of the Hamilton – Jacobi equations.
e) Separation of variables. The success of this approach depends on the
possibility of solving the above equation (13) which takes the form
n
where g
is given in (11).
j
j=1
∂S
−1
g
j
∂u
j
2
− u
=2η
j
1
To solve this equation by separation we employ an identity frequently used
by Jacobi:
If P (z)=η
1
n
j=1
n−1
z
P (uj)
U(uj)
+ η2z
n−2
+ ···+ ηnis a polynomial then
n
n
u
j
U(uj)
= η
and
1
j=1
n
uj. (17)
=
j=1
(14)
(16)
112 Various Aspects of Integrable Hamiltonian Systems
To prove this note that the left-hand side of the first relation is the sum of the
residues of
1
2πi
P (z)
dz
U(z)
which makes the relation evident. The second can be proven similarly.
We rewrite (16) using the expression (11) for g
n
j=1
U(uj)
B
j
− u
− 2η
j
1
and Bj= −4A(uj)S
j
=0.
2
u
j
Using the identities (17) this can be written in the form
n
j=1
1
U(uj)
(B
j
− u
n
j
− 2P (uj)) = 0.
Thus we can solve the equation by setting each individual term
n
B
− u
− 2P (uj)=−4A(uj)S
j
j
2
n
− u
j
j
− 2P (uj)=0.
u
Setting
Q(z)=z
n
+2η1z
n−1
+ ···+2η
n
the equation is separated into
which is solved by
S =
∂S
∂u
1
2
j
n
j=1
2
0
= −
u
j
"
Q(u
j
4A(uj)
Q(z)
− A(z)
)
dz.
Introducing the polynomial R(z)= − A(z)Q(z) the differential equations (15)
become
n
1
#
2
j=1
p
z
R(z)
z=u
du
j
= δ
dt
j
p, n−1
for p =0, 1, ..., n − 1. (18)
References 113
Clearly the η
, ..., ηnare integrals in involution and it is interesting to
1
relate them to the rational integrals derived before. For this purpose we write
n
Q(z)
A(z)
as partial fractions. It is clear that the coefficients are functions of the η
M
z −a
ν
ν
, ..., η
1
(19)
=
ν=0
and hence integrals in involution. We claim that the Mν= Fνare the previous
integrals.
We sketch the proof: Since both Q(z)/A(z) and
n
F
ν=0
ν
z − a
ν
(20)
are integrals of the motion it suffices to identify them for some point of an orbit.
For this we take the zeroes, say z = q
of Q(z). At these points one has ˙u =0
j
from (18) and hence ˙x =0and therefore the last expression becomes
n
ν=0
which vanishes precisely for z = u
tions (18) it is clear that for u
a
, ..., an, q1, ..., qnof R(z). Because of the ordering (9) it follows qj= uj.
1
Hence (19) has the same roots and poles as (20) when ˙u
2
x
z −a
1,u2
=0one has ujis equal to one of the roots a0,
j
U(z)
ν
=
ν
A(z)
, ..., un. From the differential equa-
=0. Since both ex-
j
pressions are integrals they agree up to a factor, which is 1 by comparing the
asymptotic behavior. Hence we have the identity
n
ν=0
F
z − a
Q(z)
ν
=
ν
A(z)
.
n
This formula shows that the F
are functions of η1, ..., ηn, hence integrals in
ν
involution.
References
[1] C. Neumann, De problemate quodam mechanico, quod ad primam integral-
ium ultraellipticorum classem revocatur, Journ. reine Angew. Math. 56,
1859, pp. 46–63.
114 Various Aspects of Integrable Hamiltonian Systems
[2] K. Uhlenbeck, Minimal 2-spheres and tori in Sk(informal preprint, received
1975).
[3] R. Devaney, Transversal homoclinic orbits in an integrable system,Am.
Journal Math, 100, 631–642, 1978.
On the separation of Hamilton – Jacobi equations:
[4] E. Rosochatius,¨Uber Bewegungen eines Punktes, Dissertation at Univ. Got-
tingen, Druck von Gebr. Unger, Berlin, 1877 (available at Library of the
Math. Institut, G¨ottingen).
[5] P. St¨ackel,¨Uber die Integration der Hamilton – Jacobischen Differentialgle-
ichung mittelst Separation der Variablen, Habilitationsschrift, Halle 1891
(available at Library of the Math. Institut, G¨ottingen).
§ 8. Hill’s Equation
a) In recent years remarkable progress has been achieved in the description
of the spectrum of the Hill’s equation, including the description and construction
of those periodic potentials which belong to a given spectrum, the socalled
inverse spectral problem. We can enter into this complicated and intriguing
subject only to a very limited extent. We want to show that the integrable
system of differential equations of Section is intimately connected with the Hill’s
equation in the case of a finite gap potential.
We begin with a description of the spectrum Hill’s equation and the related
inverse problem. The Hill’s equation — so called because Hill encountered it in
his lunar theory — is of the form
2
+ q(x))ϕ(x)=λϕ(x),x∈ R
(−D
1
(1)
where D = d/dx and q(x)=q(x +1) is called the potential of the operator,
which is assumed to be continuous (at least). With this operator one can associate
a number of spectra depending on the domains of definition and the boundary
condition.
Perhaps the most common problem is to consider in a dense linear manifold
2
(R1) in which the operator is essentially self adjoint. In this case the spec-
of L
trum is continuous and consists in general of infinitely many intervals extending
to +∞ («band spectrum»). In exceptional cases one has only finitely many such
intervals, one being infinitely long.
§8. Hill’s Equation 115
The end points of the above continuous spectrum are given by the operator
2
+ q(x) considered in a dense linear manifold of L
−D
2
[0, 2] of periodic
per
functions of period 2 (not 1) with the boundary condition ϕ(x +2) = ϕ(x).This
operator has a discrete spectrum
λ
0<λ1
λ2<λ3 λ4<... (2)
where every second inequality is in the strict sense. If two eigenvalues coalesce
they become double eigenvalues, like for q ≡ 0.
The spectrum for the operator on the whole line is given by the intervals
], [λ2,λ3], ...
[λ
0,λ1
which may touch each other in exceptional cases.
The third case of a boundary condition is given by considering −D
2
a dense linear manifold of L
[0, 1] with the boundary condition
2
+ q in
ϕ(0) = ϕ(1) = 0. (3)
This problem gives rise to a discrete spectrum, with simple eigenvalues μ
μ
, ...which lie in the following intervals
2
λ
μj λ2j(j =1, 2, ...);
2j−1
in particular if λ
= λ2jthen all 3 numbers λ
2j−1
, λ2j, μjagree.
2j−1
The determination of these spectra is a standard and well known application
of spectral theory. In order to describe its solution one introduces a basis
(x, λ), y2(x, λ) of (1) normalized by
y
1
y
1y2
y
1
=
y
2
10
01
at x =0.
,
1
The Floquet multipliers are the eigenvalues of the matrix
y
1y2
y
1
.
y
2
x=1
Since its determinant is 1 (by the constancy of the Wronskian) the Floquet
multipliers have a product 1 and as sum the trace of the above matrix, which is
called the discriminant.
Δ(λ)=y
(1,λ)+y
1
(1,λ).
2
116 Various Aspects of Integrable Hamiltonian Systems
In order to have a periodic solution of period 1 or 2 one needs Floquet
multipliers (+1, +1) or (−1, −1),i.e. Δ(λ)=+2or = −2. This shows that
the eigenvalues λ
, λ1, λ2, ...are given as the roots of the equation
0
2
(λ) − 4=0. (4)
Δ
Obviously the eigenvalues for the boundary condition (3) are given as roots of
the equation
(1,λ)=0; λ = μ1,μ2, ... (5)
y
2
All functions y
, y2, Δ are entire functions of order 1/2. This implies that
1
Hadamard’s factorization theorem applies and we have, for example,
∞
y
(1,λ)=c
2
j=1
μ
j
j2π
− λ
2
since μ
∼ j2π2as j →∞. This constant c is equal to 1 as one verifies from
j
the asymptotic behavior
√
λ
Thus y
(1,λ) ∼
y
2
(1,λ) is uniquely determined by the spectrum μ1, μ2, ...
2
sin
√
as λ → +∞.
λ
b) The inverse spectral problem can be formulated as follows.
(i) Which sequences λ
(ii) Given such admissible λ
, λ1, ...occur as spectra
0
, λ1, ...find and construct all potentials q(x)
0
giving rise to this spectrum
(iii) Which additional data allows us to fix q(x) uniquely.
The answer to these questions can be found in the quoted references. The
answer to (i) is contained in Marˇcenko and Ostrovskii, that to (ii) is the subject
of most of the references. As to (iii) we point out that the prescription of the
additional spectrum μ
, μ2, ... for the boundary condition (3) fixes q uniquely.
1
Moreover, one has the explicit formula
∞
+
(λ
q(0) = λ
0
j=1
+ λ2j− 2μj)(6)
2j−1
which suggest the following alternate procedure:
§8. Hill’s Equation 117
For a point ξ determine the eigenvalues μ
(ξ), μ2(ξ), ... for the boundary
1
condition
ϕ(ξ)=ϕ(ξ +1)=0
2
(ξ, ξ +1). Then the above formula gives
in L
∞
+
q(ξ)=λ
0
(λ
j=1
+ λ2j− 2μj(ξ)). (6)
2j−1
Thus giving an explicit formula for q from these various spectra.
We will treat here only the case of a finite gap potential, i. e. we assume
that λ
we will assume
= λ2jholds for all but a finite number of subscripts j. For simplicity
2j−1
λ
2j−1<λ2j
= λ2jfor j>N,
λ
2j−1
for j =1, 2, ..., N,
(7)
although the general case can be treated in the same way.
It is not even obvious that this case can occur, and less how to find the
corresponding potentials.
We will show
(A) The simple eigenvalues λ
<λ1< ... < λ2Ndetermine the double
0
eigenvalues, and hence Δ(λ) uniquely.
(B) The set of potentials q(x) belonging to a set of admissible λ
<... <λ2Nform a torus TNof dimension N;theq’s are given as
<λ
1
0
hyperelliptic functions.
N
(C) On this torus T
flow of a particle on a sphere S
the flow q(x) → q(x +t) is the same as the integrable
N
under the influence of a quadratic potential
(discussed in the previous section) when restricted to a torus obtained by fixing
N integrals F
= cj, j =0, 1, ..., N.
j
Thus, in a sense, the flow q(x) → q(x + t) for the general case can be
identified with the restriction to a torus of the corresponding mechanical problem
on an infinite dimensional sphere!
c) Proof of (A): Since Δ(λ) is a function of the order 1/2 we have
and
4 − Δ
2
(λ)=c
∞
j=0
1 −
Δ
λ
λ
j
(λ)=c
2N
= c
∞
j=1
j=0
1 −
1 −
λ
λ
j
λ
λ
j
∞
j=N +1
1 −
2
λ
λ
j
<
118 Various Aspects of Integrable Hamiltonian Systems
where λ
are the zeroes of Δ(λ) which are real and located in
j
λ
λ2j.
j
λ
2j−1
Hence for j>Nall three numbers agree and the corresponding terms cancel in
the quotient
(λ)
Δ
#
4 − Δ2(λ)
Note that the left-hand side is = ±ψ
= c
if we set
"
N
j=1
j=0
2N
1 −
1 −
λ
λ
j
.
λ
λ
j
Δ(λ)=2cosψ(λ)
and ψ(λ) is obtained as a hyperelliptic integral
⎧
⎪
⎪
⎪
⎪
ψ(λ)=
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
R(s)=
⎪
⎩
λ
1
2
0
2N
(s − λj).
j=0
N
j=1
(s − λ
ds
)
#
j
,
R(s)
This formula is due to Hochstaedt.
By adding a constant to q we can assume that λ
θ(z)=
2
)=θ(z) i. e.
2
z
N
1
2
(s − λ
j=1
0
j
so that ψ(0) = 0,andsetψ(z
We can interpret ω = θ(z) as a mapping of the upper half plane Im z>0 into the
slit domain obtained by deleting vertical slits at Re ω = θ(
=0and will normalize ψ
0
ds
)
#
.
#
λ
2j−1
)=θ(#λ2j)
(Schwarz –Christoffel’s formula).
We have to choose the parameters λ
= jπ,sothatΔ(λ
)=2cosjπ =2(−1)j. Thus we can parametrize the
2j
spectrum most effectively by choosing N positive numbers h
, λ
so that θ(#λ
j
j
)=θ(#λ2j)=
2j−1
, h2, ..., hNand
1
§8. Hill’s Equation 119
define θ(z) as the unique schlicht conformal mapping of Im z>0 onto the
domain obtained from Im w>0 by deleting the slits
Re w = ±jπ, 0 < Im w h
and such that, with z
= ±#λ2j, z
±j
±j
θ(z±j)=±πj, θ(z
,j=1, ..., N
j
= ±$λ
j
)=±πj + ih
±j
j
for j =1, 2, ..., N;andθ(0) = 0. Moreover, for large values of z we have
Q(z) ∼ z.
This shows that the 2N numbers λ
but depend only on the N parameters h
Given the spectrum, however, also the λ
eigenvalues λ
=0, λ1, ..., λ2Ndetermines λ
λ
0
(j>2N ) are uniquely determined. Consequently, the spectrum
j
, ..., λ2Ncannot be chosen arbitrarily,
1
, ..., hN.
1
, λ
, ..., λ
1
2
and all the double eigenvalues, hence the
j
as well as the other
N
discriminant Δ(λ)=2cosθ(λ). This takes care of (A).
d) Description of the potential in terms of an auxiliary spectrum. We
ask for the set of all potentials belonging to a given spectrum of the type (7). To
fix a potential one may use another spectrum, e. g. the eigenvalues belonging to
the boundary conditions (3) (actually this gives rise to 2
N
potentials in general).
In this part we will be sketchy and refer to the paper (e. g. Trubowitz [6]).
We mentioned that for any potential q(x)=q(x +1)the eigenvalues μ
for (3)
j
lie in the interval
μj λ2j.
λ
2j−1
Conversely we wish to construct a potential q for arbitrary eigenvalues μ
above interval, where the λ
are given according to (7). We can compute q(0)
j
in the
j
from (6). In order to find q(x) we subject q to the translation q(x) → q(x + t)
so that the eigenvalues μ
derive a differential equation for these functions μ
are taken into μj(t) in the above interval. We will
j
(t) and integrate them; thus
j
we can recover q(t) from the formula (6) in a unique way.
Below we will derive these differential equations which show that for increasing t the eigenvalues oscillate in the interval
μj λ
λ
2j−1
back and forth, the double valuedness coming from the unspecified sign of
√
2j
Δ2− 4. We can make the choice unique if we assign the sign of#Δ(μ)2− 4
120 Various Aspects of Integrable Hamiltonian Systems
to each point, effectively making each interval into a circle. For any specification
of μ
j
in λ
μj λ2jand a sign of#Δ(μ)2− 4 one finds a unique
2j−1
potential this way. A complete proof would require the verification that q(t) so
determined is of period 1. We forego this proof and turn to the determination of
the differential equation.
e) The differential equations for q(x) → q(x + t). If we replace q(x)
by q(x + t) we replace the parameters μ
, ..., μNby μ1(t), ..., μN(t) and
1
we ask for the differential equations describing this flow. We will write this
differential equation in two different forms: In explicit form it has the form
dμ
dt
#
2
j
−R(μj)
=
A(μj)
where A(z)=
N
(μj− z)(8)
j=1
or in the implicit form
dt
0 for p =0, 1, ..., N − 2,
j
=
1 for p = N − 1.
(9)
N
2#−R(μj)
j=1
p
μ
dμ
j
The latter follows from the former by the observation that the left-hand side is
the residue sum for the integral
p
1
2πi
z
A(z)
dz
taken over a large circle. These are the differential equations asked for under (c).
The latter form exhibits the familiar Abelian sums of the differentials
p
μ
dμ
2#−R(μ)
of the first kind on the Riemann surface defined by w
2
= −R(z).
We derive the above differential equations (see E. Trubowitz): We view the
eigenvalues μ
as functionals of q and compute the differential
j
δμ
j
2
= ϕ
δq
j
where ϕjis the normalized eigenfunction of
1
2
ϕ
dx =1.
j
0
Lϕ
= −ϕ
j
+ qϕj= μjϕj;
j
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