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 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
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
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
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:
(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.