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§6. Limit Cases, Bargmann Potentials 231
with integers j
. Of course, the ω
νk
transformation. To determine these integers we will deform the heights h
are determined only up to a unimodular
ν
k
→ 0
and verify that in that case a frequency basis is approximately given by
ω
∼ 2#α
ν
∼#α
ω
ν
so that in this limit case we can take j
dependence of the frequencies on the h
conclude that ω
= ωkform a frequency basis also for positive hk. By another
k
continuity argument we can free ourselves from the assumption that the ω
− α1(ν =1, 2, ..., n− 1) (6.2)
ν+1
− α
ν+1
1
= δνkin (6.1). Because of the continuous
νk
and the fact that the jνkare integers, we
k
(6.3)
are
ν
rationally independent.
It remains to prove (6.2) and (6.3). The latter is simple enough, since
→ 0 the potential tends to the constant potential
for h
k
and therefore
Hence
iω
= w(α
ν
w(λ) →−(α
) →−(α1− α
ν+1
q(x) → α
1
ν+1
1
− λ)
)
1/2
1/2
.
= i#α
ν+1
− α1,
proving (6.3).
In order to verify (6.2) we have to find a frequency basis for q(x) if h
small, or if the n − 1 gap lengths α
− βνtend to zero.
ν+1
are
k
For this purpose we locate the solutions of the mechanical problem (5.14)
which correspond to α
situation
Φ
(ψ,ψ)=
λ
− βν=0. Because of (5.17) we have in this limit
ν+1
n
j=1
Fj(ψ,ψ)
λ − α
b(λ)
=
a(λ)
j
=
1
λ − α
1
since the other zeros and poles cancel pairwise. Thus we have for these solutions
F
=1,F2= F3= ...= Fn=0.
1
Using the explicit form (3.9) of the F
<α
<... <αnall terms of Fnare positive, hence Fn=0implies ψn=
2
=0. Inductively, we conclude ψ
= ψ3= ...= ψn=0, ψ1= ±1. Hence the
2
vector ψ = ψ(x) corresponds to the stationary solution ψ = ±e
we sec that because of α1<
k
of (5.14).
1
232 Integrable Hamiltonian Systems and Spectral Theory
To study the solutions near this equilibrium solution we linearize (5.14).
Using δ
= ξj, j =2, ..., n as local coordinates near ψ = ±e1we find
νj
from (5.14)
¨
ξ
=(−αj+ α1)ξj,j=2, ..., n.
j
The characteristic exponents are purely imaginary, namely ±i
√
αj− α1(j =
=2, ..., n), and the stationary solution is stable. The solutions are linear
combinations of
with frequencies
of q are given by ω
√
αj− α1. On account of the formula (5.16) the frequencies
=2√αj− α1which proves (6.2) and hence Theorem 6.1.
j
exp(±i
#
αj− α1x),
As a consequence of Theorem 6.1 we see that a finite gap potential of
period l = π is characterized by integers ω
. Actually this holds true even if one
j
has infinitely many gaps. More generally, this theorem allows us to construct
finite gap potentials with prescribed frequency module.
3. Stationary solutions and their stability behavior
We s a w tha t ±e
are stationary solutions of the mechanical problem. It is
1
easily seen that the most general stationary solutions are given by the eigenvectors
of A = diag(α
, ..., αn),i.e. ψ = ±ek. To study their stability behavior
1,α2
we compute the 2(n − 1) characteristic exponents, which are
√
αk− αν; ν = k. (6.4)
±
Therefore 2(k − 1) of these exponents are real and the remaining 2(n − k) are
purely imaginary. Hence ±e
are the only stable stationary solutions and ±e
1
for k 2 has an unstable manifold W−(±ek) of dimension k −1, and a stable
manifold W
(±ek) of the dimension n − k.
+
Now it is clear that the solutions on the stable or unstable manifold approach
the stationary solution at an exponential rate for t → +∞, t →−∞, respectively.
In any event, these solutions are not quasi-periodic and we want to investigate
their meaning for the spectral problem.
For this purpose we observe that for ψ = ±e
(see (3.9)) satisfy
(ψ,ψ)=0 for j = k
F
j
(ψ,ψ)=1
F
k
, ψ=0the integrals F
k
(6.5)
k
j
§6. Limit Cases, Bargmann Potentials 233
and hence with (3.8)
(ψ,ψ)=
Φ
λ
1
λ − α
. (6.6)
k
Comparing this with (5.17) we can view this as the limit case where
β
− αν→ 0 for ν =1, 2, ..., k− 1
ν
− βν→ 0 for ν = k, ..., n.
α
ν+1
In other words, we collapse the first k − 1 bands and the last n −k gaps. In
the limit we have a half infinite slit from α
to +∞ and k − 1 points αν= β
k
for ν k − 1.
sssppppppppppppppppppppppppp pppppppppppppppppp
α
1
= β
1
α
k−1
= β
k−1
α
k
This corresponds to a Riemann surface which is a sphere with k −1 punc-
tures. The differentials of the first kind can be integrated in terms of logarithms
in this case and we will be able to write down the corresponding solutions and
potentials explicitly.
The unstable and stable manifolds W
) clearly also lie on the integral
±(ek
manifolds given by (6.5) or equivalently by (6.6). From the explicit formula (3.9)
we conclude from (6.5) similarly as before that on W
= ψ
=0, ψ
n
k+1
= ...= ψ
=0and it suffices to study ψ1, ψ2, ..., ψkin its
n
±(ek
) one has ψ
k+1
= ...=
dependence on t. In other words, the problem is reduced to the motion of the
mass-point on a (k−1)-dimensional sphere and n is replaced by k. Equivalently,
it suffices to consider the case k = n and the motion on W
(±en).
±
ν
4. The flow on the unstable manifold W
+(en
)
At first it may appear as a surprise that the exponentially decaying solution,
the stable manifold, can appear as the limit situation of almost periodic solutions.
But this phenomenon is easily illustrated with the example of the nonlinear
pendulum given by the differential equation
2
¨y = κ
sin y, (6.7)
which has the integral
G =
2
˙y
2
+ κ
2
(cos y − 1) =
˙y
2
2
− 2κ
y
2
2
sin
.
2
234 Integrable Hamiltonian Systems and Spectral Theory
The familiar phase diagram for the level lines shows that the solutions for G
positive or negative are periodic but for G =0asymptotic to the unstable
equilibrium y =0(mod 2π).
Fig. 2
It turns out that this example of the nonlinear pendulum corresponds to the
motion of the Neumann problem for n =2. In fact, in this simple case we have
the motion on the circle
2
2
+ ψ
2
=1.
ψ
1
If we parametrize this circle by the angle θ:
=sinθ, ψ2=cosθ,
ψ
1
the differential equations become
− α1)sin2θ,
2
which for y =2θ, κ
2¨θ =(α
2
= α2− α1agrees with (6.7). Then the solutions of
this equation are given by elliptic integrals, which degenerate on the stable and
unstable manifolds G =0.
In this case we have
˙
θ = ±κ sin θ, κ =
√
α2− α1.
We integrate these equations by introducing the independent variable τ =tanθ/2
go that
2
2τ
1+τ
1 − τ
1+τ
2
= ±κτ
2
2
2
=sinθ =
ψ
1
ψ2=cosθ =
=˙θ
1+τ
dτ
dt
§6. Limit Cases, Bargmann Potentials 235
or
τ = τ(0) exp(±κt).
Therefore the solutions become up to phase shift and sign change:
ψ
(t)=(coshκt)−1; ψ2(t)=tanhκt. (6.8)
1
We interpret these solutions for the spectral problem. By (5.16) and α
1
= β
they give rise to a potential
or
q(t)=2(α
q(t) − α
=
2
2
ψ
1
1
− 2κ
(cosh κt)
2
2
+ α2ψ
) − α
2
2
, κ2= α2− α1. (6.9)
2
This is the wellknown (reflectionless) potential for which the spectrum is
continuous in [α
eigenfunction ψ
=(q − λ)ϕ for λ = α
, ∞], and which has a point eigenvalue at λ = α1<α2with
2
(t) (sec (6.8)). Note that ψ2is, of course, a solution of ϕ=
1
, but ψ2is not in L2(R), and therefore λ = α2is no
2
eigenvalue.
These formulae can be generalized to arbitrary n and we can find explicit
expressions (rational in exponentials) for the solutions on W
). The corre-
±(en
sponding potentials are the Bargmann potentials. As a matter of fact, in this case
we start with the formulae for the Bargmann potential and derive the solutions
for the mechanical problem.
5. The Bargmann potentials
We make use of the explicit formula for the reflectionless potentials with prescribed negative eigenvalues −κ
2
, ..., −κ
1
2
,where0 < κ1< κ2<... <κN,
N
which were constructed by V. Bargmann. They can be written in the form
⎧
⎪
q = −2
⎪
⎪
⎪
⎨
Δ=detδ
⎪
⎪
⎪
⎪
⎩
η
j
2
d
log Δ
dx
η
+
ij
κi+ κ
= ajexp(−κjx).
iηj
i, j =1, 2, ..., N
j
(6.10)
1
Here a
exponentials η
, ..., aNare N arbitrary constants. These are rational functions of the
1
= ajexp(−κjx). The original derivation of these formulae was
j
based on the inverse theory of Gelfand and Levitan.
236 Integrable Hamiltonian Systems and Spectral Theory
There is another representation which can be derived rather directly by
inserting one eigenvalue at a time. For this purpose we represent the above q in
a different form. For details see [5].
Set
cosh(κ
=
χ
j
sinh(κ
x + δj) for j odd
j
x + δj) for j even
j
(6.11)
and define the Wronskian
= W(χ1,χ2, ..., χN).
W
N
If one writes this determinant as a polynomial in ξ
=exp(κjx + δj) and ξ
j
−1
j
one finds that the coefficients are positive and hence WN> 0. (It is for this
reason that the χ
were chosen alternatively as ch, sh). One verifies that WNis
j
related to the determinant Δ in (6.10) by
N
exp(κjx + δj)W
j=1
N
(6.12)
Δ=2
N
i>j
(κi− κj)
where
N−jr
(κj)exp(2δj); r(z)=
2
d
log Δ =
dx
2
d
log WN,
dx
Therefore
a
j
=(−1)
and we can represent q by the formula (6.10) with W
ϕ(x, k)=
W (χ
, ..., χN, exp(kx))
1,χ2
W
N
N
z −κ
i
.
z + κ
i=1
in place of Δ.Ifweset
N
i
(6.13)
where the numerator is the Wronskian of the N +1functions in the argument,
then
2
d
ϕ(x, k)=(q + k2)ϕ(x, k);
dx
in other words, we have explicit formulae for the solution of the differential
2
equation with λ = −k
ϕ(x, k) ∼
.Fork = κjone finds the asymptotic behavior
N
(k − κj)exp(kx) for x →±∞.
j=1
§6. Limit Cases, Bargmann Potentials 237
But for k = κ
these solutions are square integrable and decay for x →±∞.
j
We fin d
For k = κ
ϕ(x, κ
we can simplify the formula (6.13) to
j
) ∼ const · exp(−κjx) for x → +∞. (6.14)
j
⎧
⎪
⎪
⎪
ϕ(x, κ
⎨
⎪
⎪
⎪
⎩
j
cj=exp(δj)(−1)jκ
where the carret indicates that !χ
W (χ1, ..., !χj, ..., χN)
)=c
j
has to be omitted.
j
Now we claim that
⎧
⎪
⎪
⎪
ψ
= bjψ(x, κj); bj=
j
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
ψ0= b0ϕ(x, 0); b0=
⎪
⎩
are solutions of the differential equation
ψ
j
=(q + κ
2
)ψj,j=1, 2, ..., N (6.17)
j
which satisfy the identity
N
ψ
j=0
To prove this we note that
W
N
2
j
ν=j
κ
− κ
exp(δ
|κ
−1
j
2
j
2
ν
)
j
− κ
)
2
1/2
|
ν
(κ
j
ν=j
κ
j
N
j=1
2
=1. (6.18)
j
(6.15)
(6.16)
f(x, k)=
(k − κ
∼ exp(kx) for x →∞.
)
j
ϕ(x, k)
We can use f (x, k), f (x, −k) as solutions decaying at x → +∞, x →−∞
respectively. Their Wronskian is −2k, and we look at the function
f(x, k)f (x, −k)
k
ϕ(x, k)ϕ(x, −k)
=
(−1)Nk
N
j=1
(k2− κ
2
)
j
238 Integrable Hamiltonian Systems and Spectral Theory
which is rational in k.Fork →∞it behaves like k−1and therefore we have
1=
1
2πi
f(x, k)f (x, −k)
|k|=R
k
dk
if we integrate over a large circle. If we compute the residues at k = ±κ
and
j
use
ϕ(x, −κ
)=(−1)jexp(δj)ϕ(x, κj)
j
we obtain (6.18).
Now we interpret the formulae (6.17), (6.18) as solution of the mechanical
problem. We set
α
=0,α
n
and see that the ψ
n−j
of (6.16) are solutions of the Neumann problem. To be
j
2
= −κ
,j=1, 2, ..., N = n − 1,
j
precise the labelling has to be reversed: j → n − j. Because of (6.14) we see
that ψ
∼ const ·exp(κjx), i. e. these solutions approach the ψj= ±δjnat an
j
exponential rate. In fact, because of
=#−α
κ
j
they are precisely the characteristic exponents at e
solutions on the stable manifold W
parameters δ
they represent all solutions of W+(en).
j
=#αn− α
n−j
). Since they depend on N = n − 1
+(en
,
n−j
, and the solutions (6.16) are
n
Thus we found explicit formulae for the solutions on the stable manifold and
see that they are rational functions in the exponentials exp(κ
= n−1=1these are precisely the solutions (6.8) for δ
1
x + δj).ForN =
j
=0. The corresponding
potentials q(x) are also rational in exponentials and decay for x →±∞.
The geometrical picture shows clearly how these decaying Bargmann potentials are obtained as limit cases of quasi-periodic potentials. Geometrically
this corresponds to singularities of the foliation given by the integrals, due to the
linear dependence of the gradients of the integrals.
W
2
= χ1χ
2
2
− χ
χ2> 0
1
2
. In this case one finds
6. A focussing property on S
For a later application we write down these solutions explicitly for N =2,
i. e. for the flow on the two-dimensional sphere in R
χ
=cosh(κ1x + δ1),χ2= sinh(κ2x + δ2)
1
§6. Limit Cases, Bargmann Potentials 239
and
⎧
⎪
⎪
ψ
1
⎪
⎪
⎪
⎪
⎪
⎨
ψ2=$κ
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ψ0=
= −$κ
1
κ1κ
2
2
2
2
− κ
2
− κ
κ
2
2
χ
2
1
W
χ
1
2
1
W
χ2− κ
1
χ
2
2
2
2
χ1χ
1
2
W
2
and one verifies that these are solutions of
with
which satisfy
=(q + κ
ψ
j
2
)ψj,j=0, 1, 2; κ0=0
j
2
d
2
1
dx
log W
2
2
+ ψ
=1. (6.20)
2
q = −2
2
ψ
+ ψ
0
Moreover, for x →∞we find
ψ
→ 1,ψ1→ 0,ψ2→ 0
0
which expresses that these solutions belong to the stable manifold W
It is a remarkable geometrical fact that these orbits of W
+(e0
through another point, namely the point
2
κ
1
. (6.21)
κ
2
(ψ
0,ψ1,ψ2
)=−
κ
1
, 0, ±.1 −
κ
2
(6.19)
).
+(e0
) all pass
Indeed, when an orbit passes the plane ψ
=cosh(κ2x + δ2)=0
χ
2
and therefore
ψ
= −
0
κ1κ
2
κ
1
1
2
χ1χ
The third component follows from (6.20).
For the confocal cones
2
χ
0
λ
+
2
χ
1
λ + κ
+
2
1
=0then by (6.19) we have
1
χ1χ
λ + κ
2
κ
= −
2
2
κ
=0
1
.
2
2
2
χ
2
240 Integrable Hamiltonian Systems and Spectral Theory
one obtains a singular cone for λ = −κ
points (6.21). The observation that all orbits of the stable manifold W
2
given by the lines through the
1
+(e0
pass through the focal point was pointed out to me by R. McGehee.
7. N -solitons
It is well known that the formula (6.10) can be used to construct special
solutions of the Korteweg– de Vries equation (1.2). More precisely, if we replace
in (6.10) η
by
j
= ajexp(−κjx +4κjt3)
η
j
then the resulting potential q = q(x, t) is a solution of the KdV equation.
Analogously, one has to replace (6.11) by
j
j
for j odd
for j even
2
t)+δj.
j
with
χ
=
j
θ
j
cosh θ
sinh θ
= κj(x − 4κ
These potentials q(x, t) resolve for t →±∞into N solitons and therefore one
refers to these solutions as N-solitons. The formulae (5.16) go over into
N
2
q(x, t)=−2
j=1
2
κ
ψ
,
j
j
)
where ψ
= ψj(x, t) are defined by (6.16) with the indicated replacement.
j
We want to interpret the focusing effect on the two-sphere to show the
surprising fact that the graph of the 2-soliton
t, x, q = −2(κ
2
2
ψ
+ κ
1
1
2
2
ψ
)
2
2
contains a straight line given by
x − 4κ
2
t =const,q= −2(κ
2
2
2
− κ
2
).
1
Hence on the 2-soliton there exists a position which moves with the velocity of
the faster soliton having constant elevation.
The proof is straightforward. We determine x so that ψ =0which by (6.19)
amounts to
=sinhθ2=0
χ
2
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