This post will be something about the distribution of eigenvalues of one dimensional discrete Schrodinger operators in absolutely continuous spectrum region. Namely, given a triple and a potential function
Then these together generates a family of operators
which is given by
for
For eigenfunction equations there is an associated Schrodinger cocycle dynamics
The cocycle map
is given by
Let be it’s Lyapunov exponent (see notes 1 for definition). One way to consider the distribution of eigenvalues is to consider the finite approximation. Namely, as in Notes 7, we will restrict the operator to subinterval
and consider Dirichlet boundary condition, i.e.
Denote it by
. Let
be the eigenvalues of
Then we consider the sequence of measure
It’s standard result that as
in measure for
Here as a function of energy,
is the so-called integrated density of states (IDS). Obviously
is nondecreasing. In fact, the relation between IDS and fibered rotation number
is that
Lemma 1:
Thus by notes 3 we know, is flat on resolvent set. Namely, the support of the measure
are precisely the spectrum of
for
.
Proof: To see why we let
which a polynomial in
of degree
We set
and
Then by induction it’s easy to see that
and
Thus is an eigenvector of
if and only
Hence, if and only if
where
Then we note the following facts:
(1) For near
, it’s easy to see there is constant invariant cone field. Thus there is no rotation for
for any
and for any
.
(2) for
near
. Because
lies in resolvent set.
(3) is monotonic in
.
Thus as goes from
to
,
measures the averaged times of the vector
passing through
So as
is some doubled fibered rotation number. Since
is nondecreasing and
for
near
it’s necessary that
by Notes 3.
Now I would like to Sketch a rough proof the following main theorem of this post, which is contained in the paper ‘Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum‘ by Artur Avila, Yoram Last and Barry Simon.
Theorem 2: Assume . Then for Leb a.e.
with
, we have that
tends to be an arithematic progression. Namely, it’s going to be something like:
And is nothing other than
. The convergence is independent of
.
Remark: This theorem tells that if we look at a window around with size
, where we expect finitely many eigenvalues. Then if we rescale it by
, we will find arithematic progression as
goes to infinity. This is nothing other than the local distribution of eigenvalues in absolutely continuous spectrum region. Now let’s sketch a proof.
Proof: Note that by the proof of Lemma 1, we know that is an eigenvalue of
if and only if
. By kotani theory (see Notes 2 and 3), for a.e. E with
, we have a
map
such that
.
Thus
where . In fact, for
sufficiently large, what we need to find are these
such that
.
Assume that is a measurable continuity point for both maps
and
(see Notes 11 for definition and properties of measurable continuity points). Then it suffices to show that in some sense
as
.
In fact, replacing by
, the same arguments of Lemma1-Lemma4 give the following results:
(1) .
(2) Complexifying at
with magnitude
, we get
.
(3) Then converges to some entire function
with
.
(4) implies that
sufficiently close to
for large
.
(5) By passing to a suitable subsequence, we get that
for all as
.
(6) Thus we can write it as . By (3) we have
, which implies that
is affine.
Finally, we want to show that . We need to compute the following
The above formula lies in
Since converges to
, we only need to take care the
part. Then we have
Let . Then it’s easy to see that the above formula becomes
.
Since for , we’ve scaled by
, we get
where is the Hilbert-Schmit norm (see notes 2). This completes the proof.
Here we used the fact that . See Theorem 4 of Artur Avila and David Damanik‘s paper ‘Absolute continuity of the IDS for the almost Mathieu operator with non-critical coupling‘ for detailed information.