Commun.
Math. Phys.
146,
447-482
(1992)
Communications
IΠ
Mathematical
Physics
©
Springer-Verlag
1992
Floquet
Solutions
for the
1-Dimensional
Quasi-Periodic
Schrόdinger
Equation
L.
H.
Eliasson
Department
of
Mathematics,
Royal
Institute
of
Technology,
S-10044
Stockholm, Sweden
Received
June
10,
1991
Abstract.
We
show that
the
1-dimensional
Schrδdinger equation with
a
quasi-
periodic potential which
is
analytic
on its
hull admits
a
Floquet representation
for
almost every energy
E in the
upper part
of the
spectrum.
We
prove that
the
upper part
of the
spectrum
is
purely absolutely continuous
and
that,
for a
generic
potential,
it is a
Cantor set.
We
also
show that
for a
small potential these results
extend
to the
whole spectrum.
1.
Introduction
In
this paper
we
will consider
the
Schrόdinger equation
for
a
real quasi-periodic potential
q(ωt)
with frequency vector
ω,
and for
large
energies
E or
small potential
q. We
will study
the
existence
and
non-existence
of
Floquet
solutions
or
Block
waves,
i.e. solutions
of the
form
y(t)
=
e
kt
(p
1
(t)
+
tp
2
(t}\
where
k is a
constant
and
p
ί9
p
2
are
quasi-periodic functions with
the
frequency
co
— —
vector
ω or
—
.
We
will
also
study
the
nature
of the
spectrum
σ(<£\
where
<£
is
the
closure
of the
operator
in
the
space
L
2
(R)
of
complex square integrable
functions
on R.
We
shall assume that
g
T^-^R,
T
=
R/(2πZ),
is
analytic
in a
complex neigh-
bourhood
|Imx|
< r of
Ύ
d
,
and we
shall
use the
norm
\q\,=
sup
\q(x)\.
\lmx\<r
We
shall also assume that
ω is
diophantine,
i.e.
448 L. H.
Eliasson
for
some
τ > d — 1,
where
<n>
=
<n,ω>
is the
scalar product
in
R
d
.
LetJ?=<
-—'.neZ*
I
-half
the
frequency module
ofq-
and
let
p =
ρ(E)
be the
rotation number
of
(*).
p is a
monotone
and
continuous
function,
and
(see
[1]J.
The
connected components
of
mi(p~
l
(Jί})
are the
gaps.
So the
resolvent
set
of
3?
is the
union
of all
gaps.
A
collapsed
gap is a
point
{E} for
which
ρ(E}<=Jl
A
real number
p is
said
to be
dίophantίne
(with respect
to
Jί)
if
there exist
K
and
σ
such that
p-
neZ
d
\Q
9
2
and
it is
said
to be
rational
(with respect
to
Description
of the
Results.
We
shall
formulate
our
result
for the
matrix equation
corresponding
to
(*).
(**)
\q(ωt)-E
Theorem
A.
There
exists
a
constant
C =
C(τ,
r)
such that
if
—
oo
s<C
then
the
following
hold
for E >
E
0
(|g|
r
).
A.I.
If
p(E)
is
diophantine
or
rational,
then there
exists
a
matrix
A =
Λ(E)
in
5/(2,R)
and an
analytic matrix valued
function
7:T
d
-»G/(2,R),
also
depending
on
E,
such that
X(t)
=
Y(
—
i
A2.
If
p(E)
is
neither diophantine
nor
rational,
then
liminf
\X(t)-
X(0)\
<-\X(Q)\
and
lim
^^
= 0.
|ί|-χ»
2
|f|->oo
t
Theorem
A is a
statement about reducibility
of
equation
(**).
Indeed,
YI
—t
solves
the
equation
^
dt
\2/
\q(ωt)-E
0
for
almost every rotation
p(E)
>
p(E
0
).
Linear periodic systems
are
always reducible
as
was
shown
by
Floquet
-
Floquet theory
- but the
situation
for q-p
systems
is
more complicated.
Floquet
Solutions
for
ID
Quasi-Periodic Schrόdinger
Equation
449
In
the
resolvent
set it
is_known that
(**)
is
reducible
(see
[2]).
The first
positive
result
in the
spectrum
σ(=£?)
was
obtained
by
Dinaburg-Sinai
[3]
(see
also
[4]).
They showed
the
existence
of a set
0i\
cz
]E
0
,
oo
[nσ(J^)
such that
(**)
is
reducible
and
p(E)
is
diophantine
for all
Ee^
l
.
This
set,
however,
is not of
full
measure
in
]£
0
,
oo
[nσ(j^).
Moser-Pόschel
in [2]
constructed
a set
@
2
c
]£
0
,
oo[nσ(J^)
for
which
(**)
is
reducible
and
p(E)
is
rational
for all
Ee^
2
-
^
ut
this
set was
also
not
as
large
as one
could reasonably hope for.
In
fact,
both
3t^
and
^
2
are
defined
by
certain arithmetric conditions
on
p(E\
and
these conditions
can be
relaxed only
by
letting
E
0
become larger
-
which essen-
tially
amounts
to
require
a
stronger smallness condition.
The
principal achievement
of
this
paper
is
that
the
smallness condition
is
completely
freed
from
any
dependence
of
the
arithmetic properties
of
p(E) other than being diophantine
or
rational.
By
Theorem
Al,
(**)
is
reducible
for
a.e.
rotation number
pε{ρ(E):E>
E
0
},
but
one may ask if
this
also
holds
for
a.e.
E >
E
0
.
That this indeed
is the
case
is
the
content
of the
following corollary.
Corollary.
2p(E)p'(E)
^
1 for
almost every
Eeσ(J^)n]E
0
,
oo[.
In
particular,
the set
of
all
E>
E
0
for
which
p(E)
is
neither diophantine
nor
rational
is of
measure
0.
The
corollary
is
almost immediate.
It is
known that
2p(E)p'(E)
^
1 for
a.e.
Eε{E:γ(E)
=
0},
where γ(E)
is the
"maximal Lyapunov exponent"
of (*)
(see
[5]).
If
now
p(E)
is
diophantine, then
y(E)
= 0 by Al, and
if if
p(E)
is
neither diophantine
nor
rational, then
y(E)
= 0 by A2.
Hence γ(E)
= 0 for
a.e.
E in the
upper part
of
the
spectrum.
One may
reasonably
ask if
(**)
is
reducible
for all E >
E
0
.
There
are of
course
q
for
which this
is the
case
- q =
const
for
example
- but
this
is not the
generic
situation.
In
fact,
if X is
reducible with p(E) neither diophantine
nor
rational,
then
lim
=
0 by A2,
which implies that
X is
bounded.
The
existence
of
iπ-oo
t
unbounded such solutions
is the
content
of the
next theorem.
Theorem
B. For E >
E
0
(\q\
r
)
the
following
hold.
Bl.
The
matrix
A(E)
=
0
if{E}
is
a
collapsed
gap,
and
it
is
nilpotent
+
0 ifE
is
an
endpoint
of a
gap.
B2.
For a
generic
set of
q's,
in the
\q\
r
-topology,
there
exist
E>E
0
(\q\
r
)
for
which
X is
unbounded
and
p(E)
is
neither diophantine
nor
rational.
There
are
several examples
in the
literature
of
non-reducible linear
q-p
systems,
but
these examples
are all
non-smooth
in the
sense that
q
only
is
continuous
on
Ύ
d
.
To our
knowledge
the
only smooth examples
are [6]
(see
also
[7]).
These
examples
sit in the
bottom
of the
spectrum
and are
exponentially localized, while
the
above result concerns
the
upper part
of the
spectrum,
and the
solutions
are
clearly
not
localized because
of A2.
Theorem
C. For E >
E
0
(\q\
r
)
the
following
hold.
Cl.
For a
generic
potential
q,
in the
\q\
r
-topology,
σ(JSf)n]£
0
(|g|
Γ
),
oo[
is a
Cantor
set._
C2.
σ(JS?)
n
]
E
0
(
I
q
|
r
),
oo
[ is
purely absolutely
continuous.
In
particular,
there
are
no
point
eigenvalues
in
]£
0
(klrX
°°[
450 L. H.
Eliasson
tl
follows,
as
described
in
[2],
from
Theorem
A. And
from
A it is
also clear
that
there
is no
point spectrum
in
]£
0
(l4lr)>
°°[
(For
previous results
in
this
direction,
see
[8,9].)
In
particular,
if q is
small there
is no
point spectrum
at
all,
in
distinction
to the
case
when
q is
large
and
point spectrum
can
occur
[6].
It
was in [3]
that
the
existence
of
some absolutely
continuous_
spectrum
was
first
proven. From
[10]
we
know that
the^Lebesgue
measure
\σ
ΆC
(^)Δy~
1
(Q)\
is 0,
and
from
[1] we
know that
y'HOJczσO^.JSince
y^O
on
σ(&)
by
Theorem
A,
and
since
σ
ac
(J$?)
c
σ(&)
we
have that
\σ(&)Δσ
ac
(&)\
= 0.
Since both
σ(&)
and
σ
ac
(£f)
are
closed
we can
conclude that they
are
equal,
if we
just
know that,
for
any
interval
/,
σ(J^)n/
/
0Hσ(
J^)n/|
>
0.
This follows quite easily
from
the
estimates given below,
but it
does
not
establish
C2,
since there
may
still
be
some singular continuous spectrum.
In
order
to
rule
this
out we
shall show, following
[3],
that
all
spectral measures
are
absolutely
continuous
with respect
to the
Lebesgue measure
on the set
]£
0
(klr)»
°°[.
Let
us
also mention that some cases
of the
discrete
Schrodinger
equation
has
been shown
to
have purely absolutely continuous spectrum
[11,
12].
Idea
of
proof
. The
problem
is to
study
an
equation
where
A
1
is
constant
and
F!
is
small,
\F
1
\^ε
1
say.
This
is
obvious
if q is
small,
but it is
true
for any
q,
as was
found
by
Dinaburg-Sinai,
if E is
large enough.
One
wants
to
construct
a
transformation
Y
2
such that
Y'
2
=
(A
l
+F
1
)Y
2
-Y
2
(A
2
+
F
2
),
(1.1)
where
A
2
is
constant
and
F
2
is
much smaller than
F
l5
in
order
to
start
up an
iteration.
To do
this
one
solves
a
linear equation
Y'
2
=
lAι>Y2l
+
Fι-(A2-Aά
(1-2)
and
simply
defines
F
2
by
(1.1).
In
order
for
F
2
to be
small
one
needs
a
diophantine
condition
on the
imaginary parts
+ϊ'α
1
of the
eigenvalues
of
A±\
\2*,-(ny\^K-
l
\n\-\
rceZ
d
\0,
(1.3)
where
K±
may be
large
but not too
large.
In
fact,
K^ε^
must
be
small
but may
be
much larger than
ε
l9
so one can
take
K
1
~
s~
σ
for σ < 1, for
example.
If
this holds, then
one
gets
a
solution
of
(1.1) with
Y
2
close
to the
identity
and
A
2
close
to
A±.
And
then
one can
repeat
the
same procedure
for
A
2
+
F
2
if
just
for
some
K
2
~
|F
2
Γ
σ
.
This
is the
approach taken
by
Dinaburg-Sinai
in
[3].
One
crucial point here
is
that
one
tries
to
construct
the
transformation
Y
2
as
being close
to the
identity.
It is
this requirement which imposes
the
condition (1.3),
or at
least
a
part
of it, on
oq.
In
[2],
Moser-Poschel
studied
the
case where (1.3)
is
satisfied
(for
a
reasonable
KJ
for all rceZ
rf
\0
except
for
one. They found that
one
could still transform
A
1
+F
ί
to
A
2
+F
2
with
F
2
small,
if one
permits
a
transformation
Y
2
which
is
close,
not to the
identity,
but to an
exponential
function
e
Bt
.
(Of
course,
Y
2
will
not be a
solution
of the
linear equation (1.2)
in
this case,
and
A
2
will
not be
close
to
A
ίf
)
Floquet
Solutions
for
ID
Quasi-Periodic Schrόdinger Equation
451
So
using this idea
one can
relax
the
condition
on
α
x
in
such
a way
that
one
requires
that
the
inequality
in
(1.3)
holds
for all
integer vectors
neZ
d
\0
except
possibly
one.
Of
course, even this weaker condition
is not
always
fulfilled,
but it
is
a
well known
fact
that
it
suffices
to
require such
an
inequality
for
\n\^N
l9
where
N
1
can be
taken
to be
~log(
— 1,
since
F
l
is
analytic
and its
Fourier
\
p I
coefficients
decay exponentially.
v
1
'
Hence,
we
must require that
|2α
x
-
<^ny\^K~
1
\n\~
s
,
0<
|n|
^N
l9
except possibly
one. (1.4)
Now, (1.4)
is
always
fulfilled
because
if
|2α
1
-<π>|<K
1
-
1
|πΓ
I
and
|2α
x
-
<m>|
<K~
1
\m\-
3
9
with
I
n
I, I HI
I
^N
19
it
follows that
But
since
K
{
~
ε~
σ
and
N
1
~
log I — 1
this
is
impossible
for
ε
l
small enough.
\βι/
So
this permits
us to
always solve (1.1). Repeating this procedure gives
eventually
a
product
Y =
f]
7,
such that
with
A
constant.
But
since
Y,
is
close
to an
exponential
e
Ej
\
the
convergence
of
this
product
is
unsure unless
Bj
= 0 for all j
sufficiently
large. This
is
indeed
the
case
if the
rotation number
p is
diophantine
or
rational,
and
this
is the
whole
proof
of
Al.
Moreover, even
if the
product does
not
always converge
uniformly
on
T
d
,
it
does
converge
uniformly
on
compact intervals
in R
and,
hence, gives
a
representation
of the
solution. This provides
the
information
for
proving
A2. So
the
result
in A2, we
like
to
stress this,
is
obtained
by a
perturbation method which
is
not
absolutely convergent.
The
other results will follow
by the
same approach
but
will require
a
more
detailed description
of
a
ί
and its
dependence
on
parameters.
Outline
of
the
Paper.
In
Sect.
2 we
prove
a
basic
small divisor lemma
and in
Sect.
3
an
inductive lemma
-
these
are
standard
in
every KAM-approach.
The
set-up
is
chosen
with
the
only
aim of
getting
as
simple
and
uniform
estimates
as
possible.
This
has of
course
a
price
and the
smallness condition obtained
is
therefore
not
to
be
taken very seriously.
In
Sect.
4 we
prove
Al
and A2 as
easy consequences
of
the
inductive lemma.
In
Sect.
5 we
prove
Bl.
This requires
a
substantial amount
of
work,
but
Bl,
or
rather
its
"only
if"
part,
is
essential also
for the
proof
of B2. The
Cantor structure
of
the
spectrum
follows
from
A in the way
described
in
[2],
and we
only explain
this
without giving
any
details.
B2 is
proven
in
Sect.
6, and the
generic condition
is
that
"all
gaps
are
there." More precisely,
we
show that
on any
interval
Δ
a
]£
0
,
oo
[ in
which there
is a
dense
set of
gaps,
there exist solutions
as in B2.
The
absolute
continuity
of the
upper part
of the
spectrum
is
proven
in
Sects.
7 and
8
following
Dinaburg-Sinai.
For the
rotation number
of
(*),
or of the
matrix solution X(t)
of
(**),
and its
properties
we
refer
to
[1].
However,
in the
proof
we
must consider rotation numbers