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§4. The Schr ¨odinger Equation for Almost Periodic Potentials 211
then c
series is given by
One calls λ
the λ
finite linear combinations of the λ
=0only for a denumerable sequence λ = λνand the formal Fourier
λ
c
exp(iλνx).
λ
ν
ν
the frequencies of f and calls the smallest module over Z containing
ν
the frequency module(f) of f .Inotherwords,(f) consists of all
ν
with integer coefficients.
ν
This module is of basic importance for the following. In case f is periodic
−1
of period l this module is generated by 2πl
of the integral multiples of some ω>0 then f is periodic of period l =2πω
In all other cases
of the theory. If
(f) is dense on the real axis which leads to complications
=
(f) is generated by finitely many ω1, ω2, ..., ωdwith
, and conversely, if(f) consists
−1
integer coefficients one calls f quasiperiodic.
In the following we fix a countable frequency module
(
by
The advantage of
) the set of all almost periodic functions with this frequency module.
(
) overis to be small enough to be a separable space
, and denote
but big enough to be an algebra. The following construction will take place on
(
this fixed algebra. We will assume q = q(x) to be real valued and q ∈
).
4. The rotation number
As mentioned above Floquet theory has no analogue for almost periodic
potentials. However, the rotation number as defined in Theorem 4.1 does have
an analogue for real potentials q ∈
.
Theorem 4.5. If Im λ 0 and if ϕ = ϕ(x, λ) is a complex solution
of (4.2) satisfying
Im[ϕ, ϕ] > 0 for x =0
then
α(λ)=− lim
x→∞
x
1
x
0
Im
(t, λ)
ϕ
ϕ(t, λ)
dt 0
exists and defines as continuous function in Im λ 0 which is harmonic in
Im λ>0.
Theorem 4.6. Let λ be real, ϕ = ϕ(x, λ) a nontrivial solution of (4.2),
and N (x, λ) the number of zeroes of ϕ in [0,x].Then
.
α(λ)
π
= lim
x→∞
N(x, λ)
x
.
212 Integrable Hamiltonian Systems and Spectral Theory
Again α(λ) is a monotone increasing function. In the physics literature α/π
is referred to as the «density of states». If ν = ν(λ, a, b) denotes the number of
eigenvalues of (4.2) in [a, b] with the boundary conditions
ϕ(a)=0,ϕ(b)=0
then
lim
b−a→∞
ν(λ, a, b)
b − a
α(λ)
=
.
π
This follows readily from Theorem 4.6 and Sturm’s oscillation theorem.
Finally, the analogue of Theorem 4.3 is
Theorem 4.7. If I is a «gap» of L,i.e. ifI is an open interval in R \σ(L)
then α(λ) is a constant in I and
2α(λ) ∈
(q) for λ ∈ I.
Since in the case of period l this frequency module consists of j(2π/l) this
is the precise generalization of Theorem 4.3. But now the gaps are labelled by
a dense denumerable set, instead of the integers.
By Theorem 4.6 α(λ) is a monotone increasing function on the real axis
and therefore dα can be viewed as a measure. We have
Theorem 4.8. The spectrum σ(L) agrees with the support of the mea-
sure dα(λ).
This generalizes the characterization of the bands in the periodic case.
5. The Green’s function and a trace formula
The rotation number α(λ) and the holomorphic function w(λ) with
Im w(λ)=α(λ) can for Im λ>0 be related to the Green’s function G(x, y; λ)
of L which is defined as the kernel of the resolvent
=(L − λ)−1. (4.8)
R
λ
To represent this Green’s function we use two nontrivial solutions ψ
ψ
(x, λ) of (4.2) which are in L2(0, ∞), L2(−∞, 0) respectively. It is known
−
since Hermann Weyl [30] that ψ
, ψ−exist and are up to a factor uniquely
+
(x, λ),
+
determined. Then the Green’s function has the form
(x, λ)ψ−(y, λ)
ψ
G(x, y; λ)=G(y, x; λ)=
where [ψ
, ψ−] denotes the Wronskian.
+
+
[ψ+,ψ−]
for x>y, (4.9)
§4. The Schr ¨odinger Equation for Almost Periodic Potentials 213
Moreover, for ψ
if Im λ =0. In particular, ψ
The functions ψ
, ψ−one has the identities
+
⎧
⎪
⎪
⎪
⎪
[ψ
,ψ+]=−2i Im λ
+
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
[ψ
,ψ−]=2i Im λ
−
⎪
⎩
, ψ−have no zeros.
+
, ψ−are defined only up to a multiplicative factor, but
+
∞
x
x
|ψ−|2dx
∞
|ψ+|2dx
(4.10)
their logarithmic derivatives
ψ
(x, λ)
m
±
(x, λ)=
±
ψ±(x, λ)
are well defined.
According to G. Scharf [27] m
, m−and G(x, x; λ) are almost periodic
+
and satisfy
(x, λ),m−(x, λ),G(x, x; λ) ∈(). (4.11)
m
+
Hence the mean values of these functions exist. Since by (4.10) the solution
ϕ = ψ
satisfies the hypothesis of Theorem 4.5 we see that
−
α(λ)=−ImM (m
). (4.12)
−
This can be expressed in terms of the other functions in (4.11) because of
Proposition 4.9. For Im λ>0 we have
ROOF.
P
−M(m
)=M(m+)=M
−
2G(x, x; λ)
1
.
From (4.9) we have
1
G(x, x; λ)
= m
−
− m
+
214 Integrable Hamiltonian Systems and Spectral Theory
hence
M
G(x, x; λ)
1
= M(m
) − M(m+).
−
On the other hand
d
m
−
+ m+=
log G(x, x; λ).
dx
Since Im G(x, x; λ) > 0 the function log G is bounded; hence
M(m
)+M(m+)=0
−
which proves the statement.
Therefore, if we define
1
−1
w(λ)=−
M(G
2
(x, x; λ)) for Im λ>0(4.13)
then w(λ) is a holomorphic function in the upper half plane which by Proposition 4.9 and (4.12) satisfies
Im w(λ)=α(λ).
This quantity w(λ) is the analogue of the Floquet exponent and μ(λ)=
=exp(w(λ)) corresponds to the Floquet multiplier.
It turns out that the derivative of w can be represented as (see [12])
dw
= M(G(x, x; λ)) for Im λ>0(4.14)
dλ
which suggests to interpret this as a trace of the resolvent R
.SinceRλis not
λ
compact the ordinary trace does not exist, but we replace the integration over
the diagonal by taking the mean value and write
dw
dλ
= τ(R
).
λ
Integrating this relation we find
w(λ) − w(λ
or also for real λ, λ
α(λ) − α(λ
)=τ(log(RλR
0
:
0
) = lim
0
ε→+0
−1
)) = τ (log(I +(λ −λ0)Rλ)),
λ
0
τ(Im log(I +(λ −λ0)R
where the logarithm has to be properly defined.
λ+iε
),
§4. The Schr ¨odinger Equation for Almost Periodic Potentials 215
This expression is in close formal analogy to the spectral shift function of
scattering theory (see for example, Schrader [28]). However, in the case of the
spectral shift friction two operators are involved, and the analogy should not be
taken too seriously.
The holomorphic function w = w(λ) in Im λ>0 defined by (4.13) or (4.14)
satisfies
Re w(λ) < 0,
Im w(λ)
Im λ
> 0 for Im λ =0. (4.15)
The first inequality shows that ψ
, ψ−decay exponentially for x → +∞,
+
x →−∞, respectively. Moreover, Im w(λ)=α(λ) is the rotation number of
Theorem 4.5. Knowledge of α(λ) on the real axis determines w(λ) in the half
plane by
+∞
) − w(z2)=
w(z
1
1
π
−∞
α(λ) dλ
(λ − z1)(λ − z2)
.
In our study of finite band potentials this function w(λ) will provide an
interesting mapping of the half plane into a slit domain.
6. Connection with the KdV equation
It is surprising that the «Floquet exponent» w(λ) defined above is related
to an integrable Hamiltonian system in an infinite dimensional function space,
namely, the Korteweg– de Vries equation (1.2) mentioned in Section 1. First we
wish to show that this can be viewed as a Hamiltonian system in the space of
functions q = q(x) all of whose derivates are almost periodic, say in
(
In this space we consider two functionals F = F (q), G = G(q) which are
sufficiently smooth. We define the gradient ∇F with the help of the Frechet
derivative
d
F (q + εp)
dε
= M(∇F (q)p).
ε=0
For example, for the functional
).
one computes
F = M(aq
∇F = −2αq
2
+ bq3)
x
xx
+3bq2.
216 Integrable Hamiltonian Systems and Spectral Theory
Now we introduce the Poisson bracket
{F, G} = M∇F
d
dx
∇G,
which is a skew-symmetric form satisfying the Jacobi identity. For decaying
functions such a Poisson bracket was originally introduced by C. Gardner [10].
Corresponding to this Poisson bracket any functional, say H = H(q) gives
rise to the Hamiltonian system
∂q
∂
=
∂t
(∇H(q)).
∂x
For example, the Hamiltonian
2
3
+ q
x
H(q)=M
1
q
2
gives rise to the KdV equation
∂
q
t
(−q
=
∂x
+3q2)=−q
xx
xxx
+6qqx.
We will call any function F = F (q) an integral of the Hamiltonian H if
{F, H} =0.
It turns out that the Floquet exponent w = w(λ; q) when viewed as a functional
of q is such an integral, for any choice of λ in Im λ =0. Moreover, all these
integrals commute. This follows from
Theorem 4.10. The functional w = w(λ; q) defined by (4.13) satisfies
∇w = −G(x, x; λ) for Im λ =0
and
; q),w(λ2; q)} =0 for Im λ1=0, Im λ2=0.
{w(λ
1
The calculation can be found in [12].
In order to see the relation of w(λ; q) to the KdV equation we study its
asymptotic behavior for λ →−∞. It is well known that the Green’s function
admits an expansion of the form
G(x, x; λ) ∼
1
{1+λ
−1
G1(x)+λ−2G2(x)+...}, (4.16)
2√−λ
§4. The Schr ¨odinger Equation for Almost Periodic Potentials 217
where the G
= q/2. These coefficients G
are expressible as polynomials of q and its derivatives, e.g. G1=
j
can recursively be determined by comparison of
j
coefficients in the quadratic differential equation
− 2(q −λ)G)G −
(G
1
2
(G
− 1) = 0 (4.17)
2
which the Green’s function G = G(x, x; λ) on the diagonal always satisfies.
Indeed if we set for abbreviation α = ψ
(x, λ), β = ψ−(x, λ) and nor-
+
malize these so that [α, β]=1,then
G(x, x; λ)=αβ
G
= αβ + αβ
G− 2(q −λ)G = αβ +2αβ+ αβ− 2(q −λ)αβ =2αβ
hence (4.17) becomes
2αβα
β
1
(α
−
β + αβ)2+
2
1
2
=
1
(−[α, β]
2
2
+1)=0.
From (4.16) one obtains an asymptotic expansion for w(λ; q):
√
w(λ; q)=−
−λ{1+λ−1w1(q)+λ−2w2(q)+...},
where
= M(Wj(q)),
w
j
and W
are polynomials in q and its derivatives. Functionals of this type are
j
called «local functionals». As a corollary of theorem 4.10 we see that
{w
} =0, {wj,w(λ; q)} =0.
j,wk
An explicit calculation yields
w
1
w
2
w
3
.
.
.
1
M(q),
=
2
1
2
M(q
=
8
=
16
),
1
1
q
2
2
2
.
+ q
x
218 Integrable Hamiltonian Systems and Spectral Theory
So 16w3= H corresponds to the KdV equation. The wjrepresent the
infinitely many conservation laws of this equation.
As another consequence we see that w(λ; q) is preserved not only under the
KdV equation, corresponding to the Hamiltonian w
order Korteweg – de Vries equations corresponding to w
instance, the invariance under the w
flow corresponds to the trivial invariance
3
, but under all other higher
3
, j =1, 2, ... For
j
of w under translation q(x) → q(x + t).
§ 5. Finite Band Potentials
1. Formulation of the problem
As was mentioned previously, the spectrum for almost periodic potentials
can be exceedingly complicated and its nature is not yet understood. Even for
such a simple potential as
√
q(x)=A cos x + B cos(
it is not known whether the spectrum is a Cantor set, whether point eigenvalues
can occur. Here we will study the simplest almost periodic potentials for which
the spectrum is very simple and consists of finitely many intervals. We turn to
the inverse problem to determine all potentials having such finite band structure.
This will be achieved by reducing this question to the mechanical problem (3.11)
of Neumann.
We prescribe 2N +1real numbers
2x)
λ
<λ<...<λ
0
2N
and ask for all real potentials for which the spectrum of (4.1) is given by
σ(L)=[λ
] ∪ [λ1,λ2] ∪ ...∪ [λ2N, ∞); (5.1)
0,λ1
we refer to these N +1 intervals as «bands» and the complementary N +1
intervals as «gaps».
We reformulate the problem more precisely. For any real potential q the
Green’s function is real on the part of the real axis which belongs to the resolvent
set. Moreover, one also has Im G(x, x; λ) > 0 for Im λ>0 for any real
potential. We assume that G(x, x; λ) admits an analytic continuation to the
interior of the bands and is purely imaginary there. This requires a branch point
at the λ
of order 2.
j
§5. Finite Band Potentials 219
We make the following further assumptions. The desired q = q(x) has a
Green’s function G(x, y; λ) holomorphic in Im λ>0 such that
G(x, x; λ) is real in the gaps and purely imaginary in the bands
and behaves like
G(x, x; λ) ∼ γ
(λ − λj)
j
−1/2
near λj; γj=0. (5.3)
(5.2)
The last condition can be expressed by saying that G(x, x; λ) is a meromor-
phic function on the Riemann surface which is obtained by slitting two copies
of the complex plane along the bands and glueing the slits crosswise; moreover,
G(x, x; λ) has simple poles at λ
and at ∞. The genus of this Riemann surface
j
is N.
In addition we list the properties of G(x, x; λ) which always hold:
G(x, x; λ) is holomorphic in Im λ>0 and satisfies the differential equation
2G(G
− 2(q −λ)G) − G2+1=0, (5.4)
which we derived in (4.17), and satisfies
G(x, x; λ) ∼
1
for λ →−∞. (5.5)
2√−λ
2. Representation of G(x, x; λ) in terms of partial fractions
We rename the endpoints λ
as
j
= λ
α
j
2j−2
β
= λ
j
2j−1
j =1, 2, ..., n = N +1
j =1, 2, ..., n− 1=N
(5.6)
where N = n − 1 is the genus of the Riemann surface. With
we form
a(λ)=
n
(λ − αj),b(λ)=
j=1
"
b(λ)
−
a(λ)
n−1
(λ − βj)
j=1
.
220 Integrable Hamiltonian Systems and Spectral Theory
This is a meromorphic function on the Riemann surface. We choose that branch
in Im λ>0 which has a positive imaginary part in the bands when approaching
from above. In tho gaps this function is then real. If we form
"
−2
G(x, x; λ)=Γ(x, λ)
a(λ)
b(λ)
−
we obtain a function which is one-valued in the complex plane, has simple poles
and is regular at βj. We chose the factor −2 so that this function behaves
at α
j
−1
like λ
at λ →−∞(see (5.5)). Therefore this function is a rational function
admitting a partial fraction expansion
Γ(x, λ)=−2
.
b
−
a
G(x, x; λ)=
n
j=1
rj(x)
. (5.7)
λ − α
j
#
Moreover, since both
this function is positive in the bands [α
−b/a and G have a positive imaginary part in the bands,
], j =1, 2, ..., n−1 and [αn, ∞].
j,βj
This implies
(x) > 0 j =1, 2, ..., n. (5.8)
r
j
Proposition 5.1. If G(x, y; λ) is the Green’s function for the potential q
and if G satisfies (5.7) then there exists a real solution ψ
Lψ
= −ψ
j
+ qψj= αjψ
j
of
j
j
with
P
ROOF.
If we represent the solutions ψ
2
= ψ
r
j
j
, ψ−of the previous section as linear
+
combinations of two normalized solutions ϕ
(L − λ)ϕ =0say normalized by
then ϕ
, ϕ2are entire functions in λ and from
1
G(x, x; λ)=
ϕ
1ϕ2
ϕ
1
=
ϕ
2
x=0
[ψ+,ψ−]
.
= ϕ1(x, λ), ϕ2= ϕ2(x, λ) of
1
10
01
ψ
+ψ−
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