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§3. Geodesics on an Ellipsoid and the Mechanical System of Neumann 201
satisfying
Here Ψ
−1
B = A
(x, y) is defined like Φz(x, y) in (3.7) but with A replaced by
z
.
2
=1, q, ˙q =0, Ψ0(˙q, q)=0. (3.20)
|q|
Thus the so-parametrized geodesics on the ellipsoid correspond to the special subclass of the Neumann problem satisfying the restriction Ψ
This restriction is compatible since Ψ
(˙q, q) is an integral of the motion. Notice
0
that (3.19) differs from (3.11) in so far that A is replaced by B = A
(˙q, q)=0.
0
−1
.
The proof is rather straight forward: Differentiation of (3.15) gives
⎧
q = rBx
⎪
⎨
⎪
˙q = rB˙x +
⎩
˙r
x
r
˙r
r
= −
Bx, B ˙x
2
|Bx|
(3.21)
and from (3.16)
2
¨q = rB¨x +2˙rB ˙x +¨rBx = r(−B
= −Bq +(2˙r + rb)B ˙x +
x + bB ˙x)+2˙rB ˙x +¨rBx =
¨r
q.
r
From the expressions for b, ˙r/r in (3.17), (3.21) it follows that 2˙r + rb =0and
hence
¨q = −Bq +
¨r
q
r
which is the desired differential equation.
The mapping (x, ˙x) → (q, ˙q) defined by (3.21) is the prolongation of the
Gauss map to the tangent bundle of Q
; it is clearly a bijection.
0
A calculation shows
(˙q, q)=
Ψ
0
B ˙x, x
2
|Bx|
− 1Aq, q.
Thus the prolonged mapping takes the domain (3.18) one to one onto (3.20)
which proves the theorem.
This shows that the solutions of the geodesic problem correspond to solutions of the Neumann problem. The converse is not quite true because of the
restriction Ψ
(˙q, q)=0. While every geodesic after reparametrization satis-
0
fies (3.18) not every solution of the Neumann problem will satisfy (3.20) even
202 Integrable Hamiltonian Systems and Spectral Theory
after reparametrization. For example, the stationary solutions of (3.19) which
are given by the eigenvectors of B and ˙q =0satisfy
Ψ
(0,q)=−Aq, q < 0
0
hence violate (3.20). We want to show that anyhow all «nondegenerate» solutions
of (3.19) can be related to geodesics on a quadric
(B −μ)x, x =1.
Here we call a solution q = q(t) of (3.19) nondegenerate if
n−1
(z −μj)
(˙q, q)=
Ψ
z
j=1
det(z −B)
has a zero, say μ
the condition Ψ
, which is not an eigenvalue of B. Replacing B by B − μ1I
1
(˙q, q)=0becomes Ψ0(˙q, q)=0and the above reduction is
μ
1
possible. In short, the two problems are essentially equivalent.
Relation between the integrals
The geodesic problem (3.1) has the integrals Φ
the system (3.19) has the integrals Ψ
(˙q, q); therefore one would expect a
z
(x, y) (see (3.8)) while
z
relation between these expressions under the Gauss mapping. Kn¨orrer found the
following interesting relation: If (3.18) holds and (x, ˙x) are related to q, ˙q by
the prolonged Gauss map (3.21) then
Φ
(x, ˙x)=|Bx|4Ψw(˙q, q) where w =
z
Using the fact that y = x
= ν
1/2
˙x with ν as in (3.1) we can write this identity
1
. (3.22)
z
also in the form
Φ
(x, y)
Φ
z
(x, y)
0
(˙q, q),w= z−1,
=Ψ
w
where the prime indicates differentiation with respect to z.
To verify this relation we introduce the abbreviations
P
(p, q)=(w −B)−1p, q,Pw(q)=Pw(q, q),
w
§3. Geodesics on an Ellipsoid and the Mechanical System of Neumann 203
−1
where B = A
. With
ρ = −
˙r
=
r
Bx, B ˙x
2
|Bx|
we find from (3.21)
q = rBx, ˙q + ρq = rB ˙x
and from (3.18)
⎧
(q)=−Aq, q = −r2Bx, x = −r
P
0
⎪
⎨
P0(q, ˙q + ρq)=−Aq, ˙q + ρq = −r2Bx, ˙x =0
⎪
⎩
(˙q + ρq)=−r2B ˙x, ˙x = −1.
P
0
Moreover, the identity
(w − B)
−1
− B−1= −(z − A)−1A2for B = A−1; w = z
gives
P
(q) − P0(q)=−Qz(Aq)=−r2Qz(x)
w
(q, ˙q + ρq) −P0(q, ˙q + ρq)=−r2Qz(x, ˙x)
P
w
P
(˙q + ρq) − P0(˙q + ρq)=−r2Qz(˙x).
w
Using (3.23) we have
(q)=−r2(1 + Qz(x))
P
w
(q, ˙q + ρq)=−r2Qz(x, ˙x)
P
w
(˙q + ρq)+1=−r2Qz(˙x)
P
w
hence
P
(q)(1 + Pw(˙q + ρq)) −(Pw(q, ˙q + ρq))2=
w
4
{(1 + Qz(x))Qz(˙x) −Qz(x, ˙x)2} = r4Φz(x, ˙x).
= r
One verifies that this yields the desired identity since
2
(3.23)
−1
(˙q, q)=Pw(q)(1 + Pw(˙q)) −Pw(q, ˙q)2=Ψw(˙q + ρq, q).
Ψ
w
6. The Riemann surface
We discuss the integration of the Neumann problem (3.11) for which we
found the integrals
Φ
(˙q, q)=
z
n
k=1
Fk(˙q, q)
z −α
k
204 Integrable Hamiltonian Systems and Spectral Theory
and consider the solutions on the integral manifolds
(˙q, q)=c
F
k
k
for c1, c2, ..., cngiven, so that
n
2
=
|q|
ck=1.
k=1
Moreover, we will consider only the «general» case where the rational function
n
c
k
z −α
k=1
has n−1 distinct real roots, say β1, β2, ..., β
β
, ..., β
1
are distinct and real. This excludes only special orbits which will
n−1
k
.Inotherwords,α1, ..., αn,
n−1
be discussed in a later section. It will turn out that it is not a real restriction to
assume that
or equivalently that c
<β1<α2<... <β
α
1
> 0.
k
n−1
<αn,
Recalling the condition q, ˙q =0we consider the flow in the (n − 1)
dimensional manifold
= {q, ˙q |<q, ˙q>=0; Fk(˙q, q)=ck,k=1, 2, ..., n}.
It is customary to use elliptic coordinates μ
, μ2, ..., μ
1
for parametrization. They are defined as the zeroes of
n
2
q
j
z − α
.
j
Q
(q)=
z
j=1
Setting
n−1
m(z)=
(z −μj); a(z)=det(z −A),b(z)=
j=1
we have, because of |q| =1,
(q)=
m(z)
a(z)
.
Q
z
on the sphere
n−1
n−1
(z −βj)
j=1
§3. Geodesics on an Ellipsoid and the Mechanical System of Neumann 205
From the residues we recover
m(αj)
2
=
q
j
a(αj)
;
i. e. we recover the q
up to sign from μ1, ..., μ
j
n−1
.
To compute ˙q we use the fact that
Φ
(˙q, q)=Qz(q)(Qz(˙q)+1)− Qz(˙q, q)2= −Qz(˙q, q)
z
2
for z = μj; hence
Q
(˙q, q)=#−Φz(˙q, q)="−
z
for z = μ
j =1, 2, ..., n− 1. These constitute n − 1 linear equations for ˙q
j
which together ˙q, q =0allow us to recover ˙q. Thus μ
viewed as parameters on the manifold
.
b(z)
a(z)
, μ2, ..., μ
1
n−1
can be
For Neumann the choice of these variables was dictated because of the
possibility of separation of variables in the Hamilton – Jacobi equation. This has
been discussed before (see [23]) and we just note that the differential equations
take the implicit form
n−1
n−j−1
μ
2#−R(μk)
k=1
˙μ
k
k
= δ
for j =1, 2, ..., n− 1
j, 1
where
R(z)=a(z)b(z).
These formulae are related to the Jacobi map of the Riemann surface
2
w
= −4R(z).
This is a hyperelliptic curve of genus n − 1 with branch points at α
β
, ..., β
2
n−1,αn
. The Jacobi mapping given by
)
(μ
k,wk
n−1
k=1
(0, 0)
n−j−1
z
2#−R(z)
dz = s
j
,
1
206 Integrable Hamiltonian Systems and Spectral Theory
takes the divisor class defined byμk, 2#−R(μk)(k =1, 2, ..., n− 1) into
a point s ∈ C
n−1
/Γ where Γ denotes the period lattice of the differentials of the
first kind. Thus the differential equation becomes
= δ
˙s
j
j, 1
or sj= δ
t + sj(0)
j, 1
in these variables. In these variables we have a linear flow, which shows that
the linear structure of this integrable Hamiltonian system agrees with the linear
structure of Abel’s theorem.
In particular, we conclude that the solutions in the general case of dis-
tinct α
, βkare quasiperiodic with at most n − 1 frequencies.
j
For the algebraic aspects of this and related problems see [1], [2].
§ 4. The Schr¨odinger Equation for Almost Periodic Potentials
1. The spectral problem
In this section we discuss the problem of determining the spectrum of the
Schr¨odinger operator
L = −
on the real line −∞ <x<+∞ in case q(x) is an almost periodic function.
A special case is a periodic function q(x)=q(x + l). It is known that for
bounded q(x) there is a unique selfadjoint extension of the operator in C
the space of twice continuously differentiable functions with compact support,
and it is this extension which we mean by L. It has its spectrum on the real axis.
In particular, in the periodic case it consists, as is well known, of a generally
infinite number of intervals tending to ∞, the so-called band spectrum. The
spectrum is continuous and no point eigenvalues occur. In the almost periodic
case the spectrum can be considerably more complicated. One may have point
eigenvalues, and one can find examples for which the spectrum is a nowhere
dense Cantor set [12, 24].
In the periodic case the spectrum can be determined with the help of Floquet
theory which guarantees the existence of a nontrivial solution of the eigenvalue
equation
(L − λ)ϕ =0 or ϕ
2
d
+ q(x)(4.1)
dx
2
(R)
0
=(q −λ)ϕ (4.2)
of the form
ϕ(x, λ)=exp(w(λ)x) p(x, λ)
§4. The Schr ¨odinger Equation for Almost Periodic Potentials 207
where p(x, λ) has the same period l as q.Inotherwords
ϕ(x + l, λ)=μ(λ)ϕ(x, λ),μ=exp(w(λ)l). (4.3)
One calls μ the Floquet multiplier and ϕ a Floquet solution. In the almost
periodic case a solution of this type with almost periodic p(x, λ) generally
does not exist and an analogue of Floquet theory is not available. However
an analogue of the Floquet multiplier does exist and can be used to locate the
spectrum of L. Our goal is to define the quantity μ(λ) and its logarithm, which
has a certain similarity to the spectral phase shift of scattering theory. We will
omit the detailed existence proofs and refer to [12]. We begin with the periodic
case, and then we define the concept in the almost periodic case by analogy.
2. The periodic case
If we write the equation (4.2) as a system of first order
= A(x)y, A(x)=
y
q − λ 0
01
,y=
ϕ
,
ϕ
then the fundamental solution of this system is the matrix solution Y (x, λ) with
Y (0,λ)=I.Wehavedet Y (x, λ)=1for all x, therefore the eigenvalues
−1
of Y (l, λ) have the product one and can be denoted by μ, μ
−1
μ + μ
=trY (l, λ)(4.4)
. Moreover,
is an entire function of λ, called the discriminant Δ(λ). Clearly, by defini-
−1
tion, μ, μ
are the Floquet multipliers. For Im λ =0the function μ(λ) is
holomorphic and can be chosen so that
|μ(λ)| < 1 for Im λ>0.
One has branch points for those λ for which μ(λ)=±1, the case of a double
eigenvalue of Y (l, λ).
For real λ also
Therefore one has μ =
one speaks of the unstable (or hyperbolic) case if μ =
(or elliptic) case if μ
μ is an eigenvalue, and agrees therefore either with μ or μ−1.
μ or μμ =1for real λ. Excluding the branch points
μ = ±1, and the stable
μ =1; μ = ± 1. Equivalently one has ±Δ(λ) > 2 or
−2 < Δ(λ) < +2 in the unstable or stable case, respectively. The set
{λ ∈ R |−2 Δ(λ) 2}
208 Integrable Hamiltonian Systems and Spectral Theory
consists of a set of intervals tending to +∞ which agrees with the spectrum σ(L)
of L. Thus σ(L) consists of the closure of the stability intervals, the so-called
«bands».
The open intervals in which ±Δ(λ) > 2 are called the gaps; they belong
to the resolvent set. Thus the spectrum is determined entirely by Δ(λ),oralso
by μ(λ).SinceΔ(λ) does not permit a generalization to the almost periodic
case we will concentrate on μ = μ(λ) and give other descriptions for it.
Instead of μ(λ) we consider
1
α(λ)=Imw(λ)=
Im log μ(λ), (4.5)
l
which is harmonic for Im λ =0and determines w up to a constant. To define
the branch of the logarithm we require that α(λ) → 0 for Re λ →−∞.Inthe
following we will restrict ourselves to the upper half plane Im λ>0 and take
for μ the solution of (4.4) with |μ(λ)| < 1.
Under these circumstances we can characterize the function (4.5) as follows.
Theorem 4.1. If Im λ 0 and if ϕ = ϕ(x, λ) is a complex solution of (4.2)
satisfying
Im[ϕ, ϕ]=Im(ϕ ϕ
− ϕϕ) > 0 for x =0 (4.6)
then ϕ(x, λ) =0for x 0, and
x
α(λ)=− lim
x→∞
1
x
0
Im
ϕ
(t, λ)
ϕ(t, λ)
dt > 0. (4.7)
We will not give a proof of this statement; it is a special case of theorem 5.1
in [12]. We just point out that the Wronskian [ϕ, ϕ] is purely imaginary and
satisfies the equation
d
[ϕ, ϕ]=2i Im λ|ϕ|
dx
2
.
Hence from (4.6) one concludes that
Im[ϕ, ϕ] > 0 for x 0
hence ϕ =0for x 0. Moreover, because of
Im[ϕ, ϕ]
ϕ
−Im
=
ϕ
|ϕ|
> 0
2
one sees that α(λ) 0.
§4. The Schr ¨odinger Equation for Almost Periodic Potentials 209
From the representation (4.7) it is clear that α(λ) is a positive harmonic
function in Im λ>0, with a continuous extension to Imλ 0. We will refer to
it as the «rotation number».
Since any positive harmonic function in the upper half plane is determined
by its values on the real axis we will give an alternate characterization of α(λ)
for real λ.
Theorem 4.2. Let λ be real, and let ϕ = ϕ(x, λ) be a nontrivial real solu-
tion of (4.2).IfN (x, λ) denotes the number of zeroes of ϕ(t, λ) in 0 t x,
then
x→∞
N(x, λ)
x
;
α(λ)
π
= lim
in particular, the limit is independent of the choice of the solution ϕ(x, λ).
This formula shows that α(λ) is a monotone increasing function (not in
the strict sense). The spectrum σ(L) is characterized as the union of the closed
intervals in which α(λ) is strictly increasing and the gaps as the open intervals
of constancy. This follows from
μ = |μ|exp(iαl)
and the fact that μ is real in the gaps. Therefore we have
Theorem 4.3. In any gap the number
αl
j =
π
is a non negative integer.
Therefore the gaps can be labelled by this integer j which counts the number
of zeroes per period. For j =0we have a half-interval (−∞,λ
) where λ
0
is the bottom of the spectrum. All other gaps are bounded intervals which
may, however, collapse to a point. For example, if q =const, all gaps with
j =1, 2, ... collapse to points.
Therefore it is natural to label the bands in the following manner:
lα(λ)
=λ ∈ R | j<
b
j
π
<j+1.
These intervals are generically disjoint but in degenerate cases they may have
endpoints in common. One can show (see theorem 5.5)
0
210 Integrable Hamiltonian Systems and Spectral Theory
Theorem 4.4. In the interior of each band one has
2
)
d(α
dα
2α
dλ
=
dλ
1,
where equality holds only for q =const.
By integrating this inequality over the band b
2(λ
α
) − a2(λ) λ− λ= m(bj)
=[λ,λ] one finds
j
where m(b
) is the length of bj.Since
j
π
α(λ
)=
(j +1),α(λ
l
)=
π
j
l
we obtain the inequality
m(b
j
) <
(2j +1)
2
l
2
π
which is independent of the potential q(x)=q(x + l).
This inequality was derived in [24]; an entirely different argument was
found by J. Garnett and E. Trubowitz
1
.
3. Almost periodic potential
Since the inverse spectral problem will lead us to almost periodic functions,
we turn now to almost periodic potentials. We simply define
as the closure
of the trigonometric polynomials
f(x)=
in the uniform topology f =sup
x∈R
M(f ) = lim
cνexp(iλνx)
ν
|f(x)|.Forf ∈we define the mean value
x
1
f(t) dt
x
x→+∞
0
and associate with f a Fourier series in the usual way: We set
c
= M(f exp(−iλx));
λ
1
To appear in J. Garnett and E. Trubowitz, One-dimensional bands.
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