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Orthogonal polynomials on the unit circle New results

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ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE: NEW RESULTS
BARRY SIMON Abstract. We announce numerous new results in the theory of orthogonal polynomials on the unit circle.

1. Introduction I am completing a comprehensive look at the theory of orthogonal polynomials on the unit circle (OPUC; we’ll use OPRL for the real-line case). These two 500+-page volumes [124, 125] to appear in the same AMS series that includes Szeg? o’s celebrated 1939 book [137] contain numerous new results. Our purpose here is to discuss the most significant of these new results. Besides what we say here, some joint new results appear instead in papers with I. Nenciu [93], Totik [128], and Zlatoˇ s [130]. We also note that some of the results I discuss in this article are unpublished joint work with L. Golinskii (Section 3.2) and with Denisov (Section 4.2). Some other new results appear in [126]. Throughout, d? will denote a nontrivial (i.e., not a ?nite combination of delta functions) probability measure on ? D, the boundary of D = {z | |z | < 1}. We’ll write d?(θ) = w(θ) dθ + d?s (θ) 2π (1.1)

dθ where d?s is singular and w ∈ L1 (? D, 2 ). π Given d?, one forms the monic orthogonal polynomials, Φn (z ; d?), and orthonormal polynomials

?n (z ; d?) = If one de?nes

Φn (z ; d?) Φn L2

(1.2)

αn = ?Φn+1 (0)

(1.3)

Date : May 5, 2004. ? Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.
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B. SIMON

then the Φ’s obey a recursion relation Φn+1 (z ) = z Φn (z ) ? α ? n Φ? n (z ) where
?

(1.4)

is de?ned on degree n polynomials by
? Pn (z ) = z n Pn (1/z ?)

(1.5)

(1.4) is due to Szeg? o [137]. The cleanest proofs are in Atkinson [3] and Landau [77]. The αn are called Verblunsky coe?cients after [146]. Since Φ? n is orthogonal to Φn+1 , (1.4) implies Φn+1
2

= (1 ? |αn |2 ) Φn
n

2

(1.6) (1.7)

=
j =0

(1 ? |αj |2 )

It is a fundamental result of Verblunsky [146] that ? → {αn }∞ n=0 sets up a one-one correspondence between nontrivial probability measures and ×∞ n=0 D. A major focus in the book [124, 125] and in our new results is the view of {αn }∞ n=0 ? ? as a spectral theory problem analogous to the d association of V to the spectral measure ? dx 2 + V (x) or of Jacobi parameters to a measure in the theory of OPRL. We divide the new results in major sections: Section 2 involving relations to Szeg? o’s theorem, Section 3 to the CMV matrix, Section 4 on miscellaneous results, Section 5 on the case of periodic Verblunsky coe?cients, and Section 6 to some interesting spectral theory results in special classes of Verblunsky coe?cients. I’d like to thank P. Deift, S. Denisov, L. Golinskii, S. Khruschchev, R. Killip, I. Nenciu, P. Nevai, F. Peherstorfer, V. Totik, and A. Zlatoˇ s for useful discussions.

? ’s Theorem 2. Szego In the form ?rst given by Verblunsky [147], this says, with ? given by (1.1), that


(1 ? |αj |2 ) = exp
j =0 0



log(w(θ))

dθ 2π

(2.1)

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2.1. Szeg? o’s Theorem via Entropy. The sum rules of Killip-Simon [70] can be viewed as an OPRL analog of (2.1) so, not surprisingly, (2.1) has a “new” proof that mimics that in [70]. Interestingly enough, while the proof in [70] has an easy half that depends on semicontinuity of the entropy and a hard half (that even after simpli?cations in [129, 123] is not so short), the analog of the hard half for (2.1) follows in a few lines from Jensen’s inequality and goes back to Szeg? o in 1920 [134, 135]. Here’s how this analogous proof goes (see [124, Section 2.3] for details): (a) (well-known, goes back to Szeg? o [134, 135]). By (1.7),
n

(1 ? |αj |2 ) ≥
j =0

iθ 2 exp[log(w(θ)) + log|Φ? n (e )| ]

dθ 2π dθ 2π

(2.2) (2.3) (2.4)

≥ exp = exp
0

iθ log(w(θ)) + 2 log|Φ? n (e )| 2π

log(w(θ))

dθ 2π

dθ where (2.2) uses d? ≥ w(θ) 2 , (2.3) is Jensen’s inequality, and π ? ? (2.4) uses the fact that since Φ? n is nonvanishing in D, log|Φn (z )| is harmonic there and Φ? n (0) = 1. 2π dθ is a relative entropy and so weakly (b) The map d? → 0 log(w(θ)) 2 π upper semicontinuous in ? by a Gibbs variational principle: 2π

log(w(θ))
0

dθ = inf 2π f ∈C (? D)
f >0

f (θ) d?(θ)?1?

log(f (θ))

dθ 2π

(2.5)

(c) By a theorem of Geronimus [37], if d?n (θ) = dθ 2π |?n (eiθ )|2 (2.6)

(the Bernstein-Szeg? o approximations), then d?n → d? weakly and the Verblunsky coe?cients of d?n obey αj (d?n ) = αj (d?) j = 0, . . . , n ? 1 0 j≥n (2.7)

Therefore, by the weak upper semicontinuity of (b),
2π 2π

log(w(θ)) d? ≥ lim sup
0 n→∞ 0

? log(|?n (eiθ )|2 )

dθ 2π

(2.8)

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B. SIMON

n?1 iθ 2 ?1/2 iθ (d) Since |?n (eiθ )| = |?? |Φ? n (e )| = n (e )|, the same j =0 (1 ?|αj | ) calculation that went from (2.3) to (2.4) shows 2π

exp
0

dθ ? log(|?n (e )| ) = 2π
iθ 2

n? 1

(1 ? |αj |2 )
j =0

(2.9)

(2.4), (2.8), and (2.9) imply (2.1) and complete the sketch of this proof. We put “new” in front of this proof because it is closely related to the almost-forgotten proof of Verblunsky [147] who, without realizing he was dealing with an entropy or a Gibbs principle, used a formula close to (2.5) in his initial proof of (2.1) The interesting aspect of this entropy proof is how d?s is handled en passant — its irrelevance is hidden in (2.5). 2.2. A Higher-Order Szeg? o Theorem. (2.1) implies


|α j |2 < ∞ ?
j =0 0



log(w(θ))

dθ > ?∞ 2π

(2.10)

The following result of the same genre is proven as Theorem 2.8.1 in [124]: Theorem 2.1. For any Verblunsky coe?cients {αj }∞ j =0 ,
∞ ∞

|αj +1 ? αj |2 +
j =0 j =0

|αj |4 < ∞ ?
0



(1 ? cos(θ)) log(w(θ))

dθ > ?∞ 2π (2.11)

The proof follows the proof of (2.10) using the sum rule
∞ ∞

exp

?1 2

|α0 | ? Re(α0 ) +

2

1 2 j =0

|αj +1 ? αj |


2 j =0

(1 ? |αj |2 )e|αj | dθ 2π

2

= exp
0

(1 ? cos(θ)) log(w(θ))

(2.12)

in place of (2.1) The proof of (2.12) is similar to the proof of (2.1) sketched in Section 2.1. For details, see [124, Section 2.8]. Earlier than this work, Denisov [25] proved that when the left side of (2.11) is ?nite, then w(θ) > 0 for a.e. θ. In looking for results like (2.10), we were motivated in part by attempts of Kupin [75, 76] and Latpev et al. [78] to extend the OPUC results of Killip-Simon (see also

OPUC: NEW RESULTS

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[92]). After Theorem 2.1 appeared in a draft of [124], Denisov-Kupin [28] and Simon-Zlatos [130] discussed higher-order analogs. 2.3. Relative Szeg? o Function. In the approach to sum rules for OPRL called step-by-step, a critical role is played by the fact that if m is the m-function for a Jacobi matrix, J , and m1 is the m-function for J1 , the matrix obtained from J by removing one row and column, then Im m1 (E + i0) = |a1 m(E + i0)|2 (2.13) Im m(E + i0) The most obvious analog of the m-function for OPUC is the Carath? eodory function F (z ) = eiθ + z d?(θ) eiθ ? z (2.14)

If {αj }∞ j =0 are the Verblunsky coe?cients of d?, the analog of m1 is obtained by letting βj = αj +1 and d?1 the measure with αj (d?1 ) = βj dθ + d?1,s . and d?1 = w1 (θ) 2 π dθ For 2 -a.e. θ ∈ ? D , F (eiθ ) ≡ limr↑1 F (reiθ ) has a limit and π w(θ) = Re F (eiθ ) (2.15) Thus, as in (2.13), we are interested in Re F (eiθ )/ Re F1 (eiθ ) which, unlike (2.13), is not simply related to F (eiθ ). Rather, there is a new object (δ0 D)(z ) which we have found whose boundary values have a magnitude equal to the square root of Re F (eiθ )/ Re F1 (eiθ ). To de?ne δ0 D, we recall the Schur function, f , of d? is de?ned by F (z ) = 1 + zf (z ) 1 ? zf (z ) (2.16)

f maps D to D and (2.14)/(2.16) set up a one-one correspondence between such f ’s and probability measures on ? D. δ0 D, the relative Szeg? o function, is de?ned by (δ0 D)(z ) = 1?α ? 0 f (z ) 1 ? zf1 ρ0 1 ? zf (2.17)

where f1 is the Schur function of d?1 . One has the following: Theorem 2.2. Let d? be a nontrivial probability measure on ? D and δ0 D de?ned by (2.17). Then (i) δ0 D is analytic and nonvanishing on D. p (ii) log(δ0 D) ∈ ∩∞ p=1 H (D)

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B. SIMON
dθ -a.e. 2π

(iii) For

eiθ ∈ ? D with w(θ) = 0, w(θ) = |δ0 D(eiθ )|2 w1 (θ) (2.18)

and, in particular, log
w1 (θ )=0

w(θ) w1 (θ)

p

dθ <∞ 2π

for all p ∈ [1, ∞). 2 (iv) If ∞ j =0 |αj | < ∞, then (δ0 D)(z ) = D(z ; d?) D(z ; d?1 )

where D is the Szeg? o function. (v) If ?j (z ; d?1 ) are the OPUC for d?1 , then for z ∈ D,
n→∞

lim

?? n?1 (z ; d?1 ) = (δ0 D)(z ) ?? n (z ; d?)

For a proof, see [124, Section 2.9]. The key fact is the calculation in D that Re F (z ) |1 ? α ? 0 f |2 |1 ? zf1 |2 1 ? |z |2 |f |2 = Re F1 (z ) 1 ? |α0 |2 |1 ? zf |2 1 ? |f |2 which follows from 1 ? |z |2 |f (z )|2 Re F (z ) = 1 ? |f (z )|2 and the Schur algorithm relating f and f1 , zf1 = f ? α0 1?α ?0f (2.19)

One consequence of using δ0 D is
dθ dθ Corollary 2.3. Let d? = w(θ) 2 + d?s and dν = x(θ) 2 + dνs and π π suppose that for some N and k ,

αn+k (d?) = αn (dν ) for all n > N and that w(θ) = 0 for a.e. θ. Then, log(x(θ)/w(θ)) ∈ L1 and Φn (dν ) 2 x(θ) dθ lim = exp log n→∞ Φn+k (d?) 2 w(θ) 2π δ0 D is also central in the forthcoming paper of Simon-Zlatoˇ s [130].

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2.4. Totik’s Workshop. In [143], Totik proved the following: Theorem 2.4 (Totik [143]). Let d? be any measure on ? D with supp(d?) = ? D. Then there exists a measure dν equivalent to d? so that lim αn (dν ) = 0 (2.20)
n→∞

This is in a section on Szeg? o’s theorem because Totik’s proof uses 2 Szeg? o’s theorem. Essentially, the fact that ∞ j =0 |αj | doesn’t depend on d?s lets one control the a.c. part of the measure and changes of ∞ 2 j =0 |αj | . By redoing Totik’s estimates carefully, one can prove the stronger (see [124, Section 2.10]): Theorem 2.5. Let d? be any measure on ? D with supp(d?) = ? D. Then there exists a measure dν equivalent to d? so that for all p > 2,


|αn (dν )|p < ∞
n=0

(2.21)

It is easy to extend this to OPRL and there is also a variant for Schr¨ odinger operators; see Killip-Simon [71]. 3. The CMV Matrix One of the most interesting developments in the theory of OPUC in recent years is the discovery by Cantero, Moral, and Vel? azquez [14] 2 of a matrix realization for multiplication by z on L (? D, d?) which is of ?nite width (i.e., | χn , zχm | = 0 if |m ? n| > k for some k ; in this case, k = 2 to be compared with k = 1 for OPRL). The obvious choice for basis, {?n }∞ n=0 , yields a matrix (which [124] calls GGT after Geronimus [37], Gragg [52], and Teplyaev [139]) with two defects: If 2 ∞ the Szeg? o condition, ∞ j =0 |αj | < ∞, holds, {?n }n=0 is not a basis and Gk = ?k , z? is not unitary. In addition, the rows of G are in?nite, although the columns are ?nite, so G is not ?nite width. What CMV discovered is that if χn is obtained by orthonormalizing the sequence 1, z, z ?1 , z 2 , z ?2 , . . . , we always get a basis {χn }∞ n=0 , in which Cnm = χn , zχm (3.1) is ?ve-diagonal. The χ’s can be written in terms of the ?’s and ?? ?n+1 ?2n?1 ) and C in terms of the (indeed, χ2n = z ?n ?? 2n and χ2n?1 = z α’s. The most elegant way of doing this was also found by CMV [14]; one can write C = LM (3.2)

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B. SIMON

with

?

1 Θ1 Θ3 ...

? ? ? ?

? ? L=? ?

Θ0 Θ2 Θ4 ...

? ? ? ?

? M=? ?

(3.3) where the 1 in M is a 1 × 1 block and all Θ’s are the 2 × 2 block Θj = α ? j ρj ρj ?αj (3.4)

We let C0 denote the CMV matrix for αj ≡ 0. The CMV matrix is an analog of the Jacobi matrix for OPRL and it has many uses; since [14, 15] only presented the formalization and a very few applications, the section provides numerous new OPUC results based on the CMV matrix. 3.1. The CMV Matrix and the Szeg? o Function. If the Szeg? o condition holds, one can de?ne the Szeg? o function D(z ) = exp eiθ + z dθ log( w ( θ )) eiθ ? z 4π (3.5)

One can express D in terms of C . We use the fact, a special case of Lemma 3.2 below, that


|αj |2 < ∞ ? C ? C0 is Hilbert-Schmidt
j =0 ∞

(3.6) (3.7)

|αj | < ∞ ? C ? C0 is trace class
j =0

We also use the fact that if A is trace class, one can de?ne [43, 120] det(1 + A), and if A is Hilbert-Schmidt, det2 by det2 (1 + A) ≡ det((1 + A)e?A ) We also de?ne wn by


(3.8)

log(D(z )) = w0 +
n=1

1 2

z n wn dθ 2π

(3.9)

so wn = Here’s the result: e?inθ log(w(θ))

(3.10)

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Theorem 3.1. Suppose {αn (d?)}∞ o condition n=1 obeys the Szeg?


|α n |2 < ∞
n=0

(3.11)

Then the Szeg? o function, D, obeys for z ∈ D, ?) (1 ? z C +zw1 D(0)D(z )?1 = det2 ?0 ) e (1 ? z C where w1 = α0 ?
n=1 ∞

(3.12)

αn α ? n?1

(3.13)

If



|αn | < ∞
n=0

(3.14)

then

?) (1 ? z C ?0 ) (1 ? z C The coe?cients wn of (3.9) are given by D(0)D(z )?1 = det wn =

(3.15)

n Tr(C n ? C0 ) (3.16) n for all n ≥ 1 if (3.14) holds and for n ≥ 2 if (3.11) holds. In all cases, one has ∞ (C n )jj wn = (3.17) n j =0

? is the matrix (C ?)k = (Ck ). Remark. C The proof (given in [124, Section 4.2]) is simple: by (4.12) below, Φn can be written as a determinant of a cuto? CMV matrix, which ? ?1 gives a formula for ?? n . Since ?n → D , the cuto? matrices converge in Hilbert-Schmidt and trace norm and since det/det2 are continuous, one can take limits of the ?nite formulae. 3.2. CMV Matrices and Spectral Analysis. The results in this subsection are joint with Leonid Golinskii. The CMV matrix provides a powerful tool for the comparison of properties of two measures d?, dν on ? D if we know something about αn (dν ) as a perturbation of αn (d?). Of course, this idea is standard in OPRL and Schr¨ odinger operators. For example, Krein [2] proved a theorem of Stieltjes [132] that supp(d?) has a single non-isolated point λ if and only if the Jacobi parameters an → 0 and bn → λ by noting both statements are

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B. SIMON

equivalent to J ? λ1 being constant. Prior to results in this section, many results were proven using the GGT representation, but typically, they required lim inf n→∞ |αn | > 0 to handle the in?nite rows. Throughout this section, we let d? (resp. dν ) have Verblunsky coe?cient αn (resp. βn ) and we de?ne ρn = (1 ? |αn |2 )1/2 , σn = (1 ? |βn |)1/2 . An easy estimate using the LM factorization shows with · p the Ip trace ideal norm [43, 120]: Lemma 3.2. There exists a universal constant C so that for all 1 ≤ p ≤ ∞,
∞ 1/p

C (d?) ? C (dν )

p

≤C
n=0

|αn ? βn | + |ρn ? σn |

p

p

(3.18)

Remark. One can take C = 6. For p = ∞, the right side of (3.18) is interpreted as supn (max(|αn ? βn |, |ρn ? σn |)). This result allows one to translate the ideas of Simon-Spencer [127] to a new proof of the following result of Rakhmanov [114] (sometimes called Rakhmanov’s lemma): Theorem 3.3. If lim sup|αn | = 1, d? is purely singular. Sketch. Pick a subsequence nj so


(1 ? |αnj |)1/2 < ∞
j =0

(3.19)

Let βk = αk if k = nj and βk = αk /|αk | if k = nj . There is a ? with those values of β . It is a direct sum of ?nite limiting unitary C rank matrices since |βnj | = 1 forces L or M to have some zero matrix ? has no a.c. spectrum. elements. Thus C ? is trace class, so by the the Kato-Birman By (3.19) and (3.18), C ? C theorem for unitaries [11], C has simply a.c. spectrum. Golinskii-Nevai [49] already remarked that Rakhmanov’s lemma is an analog of [127]. For the next pair of results, the special case λn ≡ 1 are analogs of extended results of Weyl and Kato-Birman but for OPUC are new even in this case with the generality we have.
∞ ∞ ∞ ∞ and Theorem 3.4. Suppose {λn }∞ n=0 ∈ ? D , {αn }n=0 , {bn }n=0 ∈ D

(i) (ii)

βn λn ? α n → 0 ?n → 1 λn?1 λ

Then the derived sets of supp(d?) and supp(dν ) are equal, that is, up to a discrete set, supp(d?) and supp(dν ) are equal.

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∞ Theorem 3.5. Suppose {λn }∞ and αn , βn are the Verblunsky n=0 ∈ ? D dθ dθ coe?cients of d? = w(θ) 2π + d?s and dν = f (θ) 2 + dνs . Suppose that π ∞

? j ? 1| < ∞ |λj αj ? βj | + |λj +1 λ
j =0

Then {θ | w(θ) = 0} = {θ | f (θ) = 0} (up to sets of dθ/2π measure 0). The proofs (see [124, Section 4.3]) combine the estimates of Lemma 3.2 and the fact that conjugation of CMV matrices with diagonal matrices can be realized as phase changes. That supp(d?) = ? D if |αj | → 0 (special case of Theorem 3.4) is due to Geronimus [39]. Other special cases can be found in [7, 47]. [124, Section 4.3] also has results that use trial functions and CMV matrices. Trial functions are easier to use for unitary operators than for selfadjoint ones since linear variational principles for selfadjoint operators only work at the ends of the spectrum. But because ? D is curved, linear variational principles work at any point in ? D. For example, (θ0 ? ε, θ0 + ε) ∩ supp(d?) = ? if and only if Re(e?iθ0 ψ, (eiθ0 ? C )ψ ) ≥ 2 sin2 ε 2 ψ
2

for all ψ . Typical of the results one can prove using trial functions is: Theorem 3.6. Suppose there exists Nj → ∞ and kj so 1 Nj Then supp(d?) = ? D. 3.3. CMV Matrices and the Density of Zeros. A fundamental object of previous study is the density of zeros, dνn (z ; d?), de?ned to give weight k/n to a zero of Φn (z ; d?) of multiplicity k . One is interested in its limit or limit points as n → ∞. A basic di?erence from OPRL is that for OPRL, any limit point is supported on supp(d?), while limits of dνn need not be supported on ? D. Indeed, for d? = dθ/2π , dνn is a delta mass at z = 0 and [128] have found d?’s for which ?! the limit points of dνn are all measures on D As suggested by consideration of the “density of states” for Schr¨ odinger operators and OPRL (see [101, 5]), moments of the density of zeros are related to traces of powers of a truncated CMV matrix. De?ne C (n) to be the matrix obtained from the topmost n rows and
Nj

|αkj + |2 → 0
=1

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B. SIMON

leftmost n columns of C . Moreover, let dγn be the Ces` aro mean of |?j |2 d?, that is, 1 dγn (θ) = n Then: Theorem 3.7. For any k ≥ 0, z k dνn (z ) = Moreover,
n→∞ n?1

|?j (eiθ , d?)|2 d?(θ)
j =0

(3.20)

1 Tr((C (n) )k ) n

(3.21)

lim

z k dνn (z ) ?

z k dγn (z )

=0

(3.22)

Sketch. (For details, see [124, Section 8.2].) We’ll see in Theorem 4.5 that the eigenvalues of C (n) (counting geometric multiplicity) are the zeros of Φn (z ; d?) from which (3.21) is immediate. Under the CMV representation, δj corresponds to z ?j or z ?? j for suitable (see the discussion after (3.1)) so (C k )jj = and thus 1 z dγn (z ) = n
k n?1

eikθ |?j (eiθ )|2 d?(θ)

(C k )jj
j =0

(3.23)

If

< n ? 2k , ([C (n) ]k ) = (C k )

so that (3.22) follows from (3.21) and (3.23). From (3.21) and (3.18), we immediately get Corollary 3.8. If limN →∞
N →∞ 1 N N ?1 j =0 |αj

? βj | → 0, then for any k , (3.24)

lim

∞ z k [dνN (z ; {αj }∞ j =0 ) ? dνN (z ; {βj }j =0 )] = 0

One application of this is to a partially alternative proof of a theorem of Mhaskar-Sa? [86]. They start with an easy argument that uses a theorem of Nevai-Totik [96] and the fact that (?1)n+1 α ? n?1 is the product of zeros of Φn (z ) to prove

OPUC: NEW RESULTS

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Lemma 3.9. Let A = lim sup |αn |1/n and pick nj so |αnj ?1 |1/nj ?1 → A Then any limit points of dνnj is supported on {z | |z | = A}. (3.26) (3.25)

They then use potential theory to prove the following, which can be proven instead using the CMV matrix: Theorem 3.10 (Mhaskar-Sa? [86]). Suppose (3.25) and (3.26) hold and that either A < 1 or 1 lim n→∞ N
n?1

|α j | = 0
j =0

(3.27)

Then dνnj converges weakly to the uniform measure on the set {z | |z | = A}. Sketch of New Proof. Since dθ/2π is the unique measure with δk0 for k ≥ 0, it su?ces to show that for k ≥ 1, z k dνnj → 0 This is immediate from Corollary 3.8 and the fact that ν ?n is the zero’s measure for dθ/2π . z k dν ?n = 0 if
dθ zk 2 = π

3.4. CMV and Wave operators. In [125, Section 10.7], we prove the following:
2 Theorem 3.11. Suppose ∞ n=0 |αn | < ∞. Let C be the CMV matrix ∞ for {αn }n=0 and C0 the CMV matrix for αj ≡ 0. Then n→±∞ ?n s-lim C n C0 = ?±

exists and its range is χS (C ) where S is a set with d?s (S ) = 0, |? D\S | = 0. The proof depends on ?nding an explicit formula for ?± (in terms of D(z ), the Szeg? o function); equivalently, from the fact that in a suitable sense, C has no dispersion. The surprise is that one only needs ∞ ∞ 2 n=0 |αn | < ∞. Some insight can be obtained n=0 |αn | < ∞, not from the formulae Geronimus [38] found mapping to a Jacobi matrix when the α’s are real. The corresponding a’s and b’s have the form cn+1 ? cn + dn where dn ∈ 1 and cn ∈ 2 , so there are expected to be modi?ed wave operators with ?nite modi?cations since cn+1 ? cn is conditionally summable.

14

B. SIMON

Simultaneous with our discovery of Theorem 3.11, Denisov [27] found a similar result for Dirac operators. 3.5. The Resolvent of the CMV Matrix. I have found an explicit 1 formula for the resolvent of the CMV matrix (C ? z )? k when z ∈ D (and for some suitable limits as z → ? D), not unrelated to a formula for the resolvent of the GGT matrix found by Geronimo-Teplyaev [36] (see also [34, 35]). Just as the CMV basis, χn , is the result of applying Gram-Schmidt to orthonormalize {1, z, z ?1 , z 2 , z ?2 , . . . }, the alternate CMV basis, xn , is what we get by orthonormalizing {1, z ?1 , z, z ?2 , z 2 , . . . }. (One can ? = x , zx = ML.) Similarly, let yn , Υn be the CMV and show C ? ). De?ne alternate CMV bases associated to (ψn , ?ψn pn = yn + F (z )xn πn = Υn + F (z )χn Then Theorem 3.12. We have that for z ∈ D, [(C ? z )?1 ]k = (2z )?1 χ (z )pk (z ) k > or k = = 2n ? 1 (3.30) (2z )?1 π (z )xk (z ) > k or k = = 2n (3.28) (3.29)

This is proven in [124, Section 4.4]. It can be used to prove Khrushchev’s formula [68] that the Schur function for |?n |2 d? is ?1 ∞ ?n (?? n ) f (z ; {αn+j }j =0 ); see [125, Section 9.2]. 3.6. Rank Two Perturbations and CMV Matrices. We have uncovered some remarkably simple formulae for ?nite rank perturbations of unitaries. If U and V are unitary so U ? = V ? for ? ∈ Ran(1 ? P ) where P is a ?nite-dimensional orthogonal projection, then there is a unitary Λ = P H → P H so that V = U (1 ? P ) + U ΛP V V U G0 (z ) = P U G(z ) = P G(z ) = 1 + zg (z ) 1 ? zg (z ) +z P ?z +z P ?z 1 + zg0 (z ) 1 ? zg0 (z ) (3.31) For z ∈ D, de?ne G0 (z ), G(z ), g0 (z ), g (z ) mapping P H to P H by (3.32) (3.33) (3.34)

G0 (z ) =

OPUC: NEW RESULTS

15

As operators on P H, g (z ) < 1, g0 (z ) < 1 on D. A direct calculation (see [124, Section 4.5]) proves that g (z ) = Λ?1 g0 (z ) (3.35) This can be used to provide, via a rank two decoupling of a CMV 1 0 matrix (change a Θ(α) to ( ? 0 1 )), new proofs of Geronimus’ theorem and of Khrushchev’s formula; see [124, Section 4.5]. 3.7. Extended and Periodized CMV Matrices. The CMV matrix is de?ned on 2 ({0, 1, . . . }). It is natural to de?ne an extended CMV 2 ? matrix associated to {αj }∞ (Z) by extending L and M to L j =?∞ on ? on 2 (Z) as direct sums of Θ’s and letting E = L ?M ?. and M This is an analog of whole-line discrete Schr¨ odinger operators. It is useful in the study of OPUC with ergodic Verblunsky coe?cients as well as a natural object in its own right. [124, 125] have numerous results about this subject introduced here for the ?rst time. If {αj }∞ j =?∞ is periodic of period p, E commutes with translations and so is a direct integral of p × p periodized CMV matrices depending on β ∈ ? D: essentially to restrictions of E to sequences in ∞ with un+kp = β k un . In [125, Section 12.1], these are linked to Floquet theory and to the discriminant, as discussed below in Section 5.1.

4. Miscellaneous Results In this section, we discuss a number of results that don’t ?t into the themes of the prior sections and don’t involve explicit classes of Verblunsky coe?cients, the subject of the ?nal two sections. 4.1. Jitomirskaya-Last Inequalities. In a fundamental paper intended to understand the subordinacy results of Gilbert-Pearson [42] and extend the theory to understand Hausdor? dimensionality, Jitomirskaya-Last [64, 65] proved some basic inequalities about singularities of the m-function as energy approaches the spectrum. In [125, Section 10.8], we prove an analog of their result for OPUC. First, we need some notation. ψ denotes the second polynomial, that is, the OPUC with sign ?ipped αj ’s. For x ∈ [0, ∞), let [x] be the integral part of x and de?ne for a sequence a:
[x]

a We prove

2 x

=
j =0

|aj |2 + (x ? [x])|aj +1 |2

(4.1)

16

B. SIMON

Theorem 4.1. For z ∈ ? D and r ∈ [0, 1), de?ne x(r) to be the unique solution of √ (1 ? r) ? (z ) x(r) ψ (z ) x(r) = 2 (4.2) Then ψ (z ) x(r) ψ (z ) x(r) A?1 ≤ |F (rz )| ≤ A (4.3) ? (z ) x(r) ? (z ) x(r) where A is a universal constant in (1, ∞). Remark. One can take A = 6.65; no attempt was made to optimize A. This result allows one to extend the Gilbert-Pearson subordinacy theory [42] to OPUC. Such an extension was accomplished by GolinskiiNevai [49] under an extra assumption that lim sup |αn | < 1 (4.4) We do not need this assumption, but the reason is subtle as we now explain. ? Solutions of (1.4) and its ? viewed as an equation for ? are given ? by a transfer matrix Tn (z ) = A(αn?1 , z )A(αn?2 , z ) . . . A(α0 , z ) where ρ = (1 ? |α| )
2 1/2

(4.5)

and z ?α ? ?αz 1 (4.6)

A(α, z ) = ρ?1

In the discrete Schr¨ odinger case, the transfer matrix is a product of v ?1 A(v, e) = ( e? ). A key role in the proof in [65] is that A(v, e ) ? 1 0 A(v, e) depends only on e and e and not on v . For OPUC, the A has the form (4.6). [49] requires (4.4) because A(α, z ) ? A(α, z ) has a ρ?1 divergence, and (4.4) controlled that. The key to avoiding (4.4) is to note that A(α, z ) ? A(α, w) = (1 ? z ?1 w)A(α, z )P 0 where P = ( 1 0 0 ). 4.2. Isolated Pure Points. Part of this section is joint work with S. Denisov. These results extend beyond the unit circle. We’ll be interested in general measures on C with nontrivial probability measures |z |j d?(z ) < ∞ (4.7)

for all j = 0, 1, 2, . . . . In that case, one can de?ne monic orthogonal polynomials Φn (z ), n = 0, 1, 2, . . . . Recall the following theorem of Fej? er [30]:

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17

Theorem 4.2 (Fej? er [30]). All the zeros of Φn lie in the convex hull of supp(d?). We remark that this theorem has an operator theoretic interpretation. If Mz is the operator of multiplication by z on L2 (C, d?), and if n?1 Pn is the projection onto the span of {z j }j =0 , then we’ll see (4.10) that the eigenvalues of Pn Mz Pn are precisely the zeros of Φn . If η ( · ) denotes numerical range, η (Mz ) is the convex hull of supp(d?), so Fej? er’s theorem follows from η (Pn Mz Pn ) ? η (Mz ) and the fact that eigenvalues lie in the numerical range. [124, Section 1.7] contains the following result I proved with Denisov: Theorem 4.3. Let ? obey (4.7) and suppose z0 is an isolated point of supp(d?). De?ne Γ = supp(d?)\{z0 } and ch(Γ), the convex hull of Γ. Suppose δ ≡ dist(z0 , ch(Γ)) > 0. Then Φn has at most one zero in {z | |z ? z0 | < δ/3}. Remarks. 1. In case supp(d?) ? ? D, any isolated point has δ > 0. Indeed, if d = dist(z0 , Γ), δ ≥ d2 /2 and so, Theorem 4.3 says that there is at most one zero in the circle of radius d2 /6. ] ∪ {0} ∪ [ 1 , 1] and symmetric under 2. If d? is a measure on [?1, ? 1 2 2 x, and ?({0}) > 0, it can be easily shown that P2n (x) has two zeros near 0 for n large. Thus, for a result like Theorem 4.3, it is not enough that z0 be an isolated point of supp(d?); note in this example that 0 is in the convex hull of supp(d?)\{0}. The other side of this picture is the following result proven in [124, Section 8.1] using potential theoretic ideas of the sort exposed in [116, 131]: Theorem 4.4. Let ? be a nontrivial probability measure on ? D and let z0 be an isolated point of supp(d?). Then there exist C > 0, a > 0, and a zero zn of Φn (z ; d?) so that |zn ? z0 | ≤ Ce?a|n| (4.8)

There is an explicit formula for a in terms of the equilibrium potential for supp(d?) at z0 . The pair of theorems in this section shows that any isolated mass point, z0 , of d? on ? D has exactly one zero near z0 for n large. 4.3. Determinant Theorem. It is a well-known fact that if J (n) is the n × n truncated Jacobi matrix and Pn the monic polynomial associated to J , then Pn (x) = det(x ? J (n) ) (4.9)

18

B. SIMON

The usual proofs of (4.9) use the selfadjointness of J (n) but there is a generalization to OPs for measures on C: Theorem 4.5. Let d? be a measure on C obeying (4.7). Let Pn be ?1 the projection onto the span of {z j }n j =0 , Mz be multiplication by z , and M (n) = Pn Mz Pn . Then Φn (z ) = det(z ? M (n) ) (4.10) Sketch. Suppose z0 is an eigenvalue of M (n) . Then there exists Q, a polynomial of degree at most n ? 1, so Pn (z ? z0 )Q(z ) = 0. Since Φn is up to a constant, the only polynomial, S , of degree n with Pn (S ) = 0, we see (z ? z0 )Q(z ) = cΦn (z ) (4.11) It follows that Φn (z0 ) = 0, and conversely, if Φn (z0 ) = 0, Φn (z )/(z ? z0 ) ≡ Q provides an eigenfunction. Thus, the eigenvalues of M (n) are exactly the zeros of Φn . This proves (4.10) if Φn has simple zeros. In general, by perturbing d?, we can get Φn as a limit of other Φn ’s with simple zeros. In the case of ? D, z is unitary on L2 (? D, d?), so Pn in de?ning M (n) ?1 can be replaced by the projection onto the span of {z j + }n j =0 for any , in particular, the span onto the ?rst n of 1, z, z ?1 , z 2 , . . . , so Corollary 4.6. If C (n) is the truncated n × n CMV matrix, then Φn (z ) = det(z ? C (n) ) (4.12)

4.4. Geronimus’ Theorem and Taylor Series. Given a Schur function, that is, f mapping D to D analytically, one de?nes γ0 and f1 by f (z ) = γ0 + zf1 (z ) 1+γ ?0 zf1 (z ) (4.13)

so γ0 = f (0) and f1 is either a new Schur function or a constant in ? D. The later combines the fact that ω → (γ0 + ω )/(1 + γ ?0 ω ) is a bijection of D to D and the Schur lemma that if g is a Schur function with g (0) = 0, then g (z )z ?1 is also a Schur function. If one iterates, one gets either a ?nite sequence γ0 , . . . , γn?1 ∈ Dn and γn ∈ ? D or an ∞ in?nite sequence {γj }∞ j =0 ∈ D . It is a theorem of Schur that this sets up a one-one correspondence between the Schur functions and such γ sequences. The ?nite sequences correspond to ?nite Blaschke products. In 1944, Geronimus proved Theorem 4.7 (Geronimus’ Theorem [37]). Let d? be a nontrivial probability measure on ? D with Verblunsky coe?cients {αj }∞ j =0 . Let f be

OPUC: NEW RESULTS

19

the Schur function associated to d? by (2.14)/ (2.16) and let {γn }∞ n=0 be its Schur parameters. Then γn = αn (4.14)

[124] has several new proofs of this theorem (see [44, 111, 67] for other proofs, some of them also discussed in [124]). We want to describe here one proof that is really elementary and should have been found in 1935! Indeed, it is obvious to anyone who knows Schur’s paper [117] and Verblunsky [146] — but apparently Verblunsky didn’t absorb that part of Schur’s work, and Verblunsky’s paper seems to have been widely unknown and unappreciated! This new proof depends on writing the Taylor coe?cients of F (z ) in terms of the α’s and the γ ’s. Since eiθ + z =1+2 e?inθ z n eiθ ? z n=1 we have F (z ) = 1 + 2
n=1 ∞ ∞

cn z n

(4.15)

with cn given by cn = e?inθ d?(θ) (4.16)

n De?ne sn (f ) by f (z ) = ∞ n=0 sn (f )z . Then Schur [117] noted that (1 + γ ?0 zf1 )f = γ0 + zf1 implies n

sn (f ) = (1 ? |γ0 |2 )sn?1 (f1 ) ? γ ?0
j =1

sj (f )sn?1?j (f1 )

so that, by induction,
n?1

sn (f ) =
j =0

(1 ? |γj |2 )γn + rn (γ0 , γ ?0 , . . . , γn?1 , γ ?n?1 )

with rn a polynomial. This formula is in Schur [117]. Since n F (z ) = 1 + 2 ∞ n=1 (zf ) , we ?nd that cn = sn?1 (f ) + polynomial in (s0 (f ), . . . , sn?1 (f )), and thus
n?2

cn (f ) =
j =0

(1 ? |γj |2 )γn?1 + r ?n?1 (γ0 , γ ?0 , . . . , γn?2 , γ ?n?2 )

(4.17)

for a suitable polynomial r ?n?1 .

20

B. SIMON

On the other hand, Verblunsky [146] had the formula relating his parameters and cn (f ):
n?2

cn (f ) =
j =0

(1 ? |αn |2 )αn?1 + q ?n?1 (α0 , α ? 0 , . . . , αn?2 , α ? n?2 )

(4.18)

For Verblunsky, (4.18) was actually the de?nition of αn?1 , that is, he showed (as did Akhiezer-Krein [1]) that, given c0 , . . . , cn?1 , the set of allowed cn ’s for a positive Toeplitz determinant is a circle of radius ?1 2 inductively given by n j =0 (1?|αj | ), which led him to de?ne parameters αn?1 . On the other hand, it is a few lines to go from the Szeg? o recursion (1.4) to (4.18). For we note that Φn+1 (z ) d?(z ) = 1, Φn+1 = 0 while 1, Φ? n by (1.7). Thus
n?1 n?1

= Φn , z

n

= Φn , Φn =
j =1

(1 ? |αj |2 )

1, z Φn = α ?n
j =1

(1 ? |αj |2 )

(4.19)

But since z Φn = z

n+1

+ lower order, (4.20)

1, z Φn = c ?n+1 + polynomial in (c0 , c1 , . . . , cn , c ?1 , . . . , c ?n )

This plus induction implies (4.18). n?1 Given (4.17) and (4.18) plus the theorem of Schur that any {γj }j =0 ?1 in Dn is allowed, and the theorem of Verblunsky that any {αj }n j =0 in n D is allowed, we get (4.14) inductively. 4.5. Improved Exponential Decay Estimates. In [96], NevaiTotik proved that lim sup |αn |1/n = A < 1 ? d?s = 0 and D?1 (z ) is analytic in {z | |z | < A?1 } (4.21) providing a formula for the exact rate of exponential decay in terms of properties of D?1 . By analyzing their proof carefully, [124, Section 7.2] re?nes this to prove Theorem 4.8. Suppose
n→∞ n→∞

lim |αn |1/n = A < 1

(4.22)

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21

and de?ne


S (z ) =
n=0

αn z n

(4.23)

Then S (z ) + D(1/z ?) D(z ) |z | < A?2 }.

?1

has an analytic continuation to {z | A <

The point of this theorem is that both S (z ) and D(1/z ?) D(z )?1 have singularities on the circle of radius A?1 (S by (4.22) and D?1 by (4.21)), so the fact that the combination has the continuation is a strong statement. Theorem 4.8 comes from the same formula that Nevai-Totik [96] use, 2 1/2 namely, if d?s = 0 and κ∞ = ∞ , then n=0 (1 ? |αn | ) α n = ? κ∞ Φn+1 (eiθ ) D(eiθ )?1 d?(θ) (4.24)

We combine this with an estimate of Geronimus [39] that ?? n+1 ?D
?1 L2 (? ,d?)

D







1/2

2
j =n+1

|αj |

2

(4.25)

? dθ to get and D?1 d? = D 2π


αn +
j =n

?j ?n,1 = O((A?1 ? ε)?2n ) dj,?1 d

(4.26)

j ?1 j where D(z ) = ∞ = ∞ j =0 dj,1 z , D (z ) j =0 dj,?1 z . (4.26) is equivalent to analyticity of S (z ) + D(1/z ?) D(z )?1 in the stated region. One consequence of Theorem 4.8 is

Corollary 4.9. Let b ∈ D. Then αn+1 = b + O (δ n ) αn

(4.27)

for some δ < 1 if and only if D?1 (z ) is meromorphic in {z | |z | < |b|?1 + δ } for some δ and D(z )?1 has only a single pole at z = 1/b in this disk. This result is not new; it is proven by other means in Barrios-L? opezSa? [8]. Our approach leads to a re?ned form of (4.27), namely, αn = ?Cbn + O((bδ )n ) with C= lim (1 ? zb)D(z )?1 D(? b) ?1 (4.28) (4.29)

z →b

22

B. SIMON

One can get more. If D(z )?1 is meromorphic in {z | |z | < A?2 }, one gets an asymptotic expansion of αn of the form αn =
j =1 n Pmj (n)zj + O((A?2 ? ε)?n )

where the zj are the poles of D?1 in {z | |z | < A?2 } and Pmj are polynomials of degree mj = the order of the pole at mj . There are also results relating asymptotics of αn of the form αn = Cbn nk (1 + o(1)) to asymptotics of dn,?1 or the form dn,?1 = C1 bn nk (1 + o(1)). 4.6. Rakhmanov’s Theorem on an Arc with Eigenvalues in the Gap. Rakhmanov [113] proved a theorem that if (1.1) holds with w(θ) = 0 for a.e. θ, then limn→∞ |αn | = 0 (see also [114, 84, 95, 68]). In [125, Section 13.4], we prove the following new result related to this. De?ne for a ∈ (0, 1) and λ ∈ ? D Γa,λ = {z ∈ ? D | arg(λz ) > 2 arcsin(a)} (4.30) and ess supp(d?) of a measure as points z0 with {z | |z ? z0 | < ε} ∩ supp(d?) an in?nite set for all ε > 0. Then Theorem 4.10. Let d? be given by (1.1) so that ess supp(d?) = Γa,λ and w(θ) > 0 for a.e. eiθ ∈ Γa,λ . Then
n→∞

lim |αn (d?)| = a

n→∞

lim αn+1 (d?) αn (d?) = a2 λ

(4.31)

We note that, by rotation invariance, one need only look at λ = 1. Γa ∪ {1} is known (Geronimus [40, 41]; see also [45, 46, 50, 51, 107, 108, 109, 110]) to be exactly the spectrum for αn ≡ a and the spectrum on Γa is purely a.c. with w(θ) > 0 on Γint a . Theorem 4.10 can be viewed as a combination of two previous extensions of Rakhmanov’s theorem. First, Bello-L? opez [7] proved (4.31) if ess supp(d?) = Γa,λ is replaced by supp(d?) = Γa,λ . Second, Denisov [26] proved an analog of Rakhmanov’s theorem for OPRL. By the mapping of measures on ? D to measures on [?2, 2] due to Szeg? o [136] and the mapping of Jacobi coe?cients to Verblunsky coe?cients due to Geronimus [38], Rakhmanov’s theorem immediately implies that if a Jacobi matrix has supp(dγ ) = [?2, 2] and dγ = f (E ) + dγs with f (E ) > 0 on [?2, 2], then an → 1 and bn → 0. What Denisov [26] did is extend this result to only require ess supp(dγ ) = [?2, 2]. Thus, Theorem 4.10 is essentially a synthesis of the Bello-L? opez [7] and Denisov [26] results. One di?culty in such a synthesis is that Denisov relies on Sturm oscillation theorems and such a theorem does not seem to be applicable for OPUC. Fortunately, Nevai-Totik [97]

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23

have provided an alternate approach to Denisov’s result using variational principles, and their approach — albeit with some extra complications — allows one to prove Theorem 4.10. The details are in [125, Section 13.4]. 4.7. A Birman-Schwinger Principle for OPUC. Almost all quantitative results on the number of discrete eigenvalues for Schr¨ odinger operators and OPRL depend on a counting principle of Birman [10] and Schwinger [118]. In [125, Section 10.15], we have found an analog for OPUC by using a Cayley transform and applying the BirmanSchwinger idea to it. Because of the need to use a point in ? D about which to base the Cayley transform, the constants that arise are not universal. Still, the method allows the proof of perturbation results like the following from [125, Section 12.2]: Theorem 4.11. Suppose d? has Verblunsky coe?cients αj and there exists βj with βj +p = βj for some p and


j |αj ? βj | < ∞
j =0

(4.32)

Then d? has an essential support whose complement has at most p gaps, and each gap has only ?nitely many mass points. Theorem 4.12. Suppose α and β are as in Theorem 4.11, but (4.32) is replaced by


|αj ? βj |p < ∞
j =0

(4.33)

for some p ≥ 1. Then dist(zj , ess sup(d?))q < ∞
zj = mass points in gaps

(4.34)

where q >

1 2

if p = 1 and q ≥ p ?

1 2

if p > 1.

Theorem 4.11 is a bound of Bargmann type [6], while Theorem 4.12 is of Lieb-Thirring type [81]. We have not succeeded in proving q = 1 2 for p = 1 whose analog is known for Schr¨ odinger operators [148, 62] and OPRL [63]. 4.8. Rotation Number for OPUC. Rotation numbers and their connection to the density of states have been an important tool in the theory of Schr¨ odinger operators and OPRL (see Johnson-Moser [66]). Their analog for OPUC has a twist, as seen from the following theorem from [124, Section 8.3]:

24

B. SIMON

Theorem 4.13. arg(Φn (eiθ )) is monotone increasing in θ on ? D and de?nes a measure d arg(Φn (eiθ ))/dθ of total mass 2πn. If the density of zeros dνn has a limit dν supported on ? D, then 1 d arg(Φn (eiθ )) 1 1 dθ → dν + 2πn dθ 2 2 2π weakly.
dθ is surprising. In a sense, it comes Given the OPRL result, the 1 2 2π from the fact that the transfer matrix obeys det(Tn ) = z n rather than determinant 1. The proof of Theorem 4.13 comes from an exact result that in turn comes from looking at arg(eiθ ? z0 ) for z0 ∈ D:

(4.35)

1 d arg(Φn (eiθ )) 1 1 dθ = P (dνn ) + (4.36) 2πn dθ 2 2 2π where P is the dual of Poisson kernel viewed as a map of C (? D) to C (D), that is, P (dγ ) = 1 2π 1 ? | r |2 dγ (reiθ ) 2 1 + r ? 2r cos θ (4.37)

5. Periodic Verblunsky Coefficients In this section, we describe some new results/approaches for Verblunsky coe?cients {αn }∞ n=0 that obey αn+p = αn (5.1) for some p. We’ll normally suppose p is even. If it is not, one can use the fact that (α0 , 0, α1 , 0, α2 , 0, . . . ) is the Verblunsky coe?cients 1 of the measure 2 d?(e2iθ ) and it has (5.1) with p even, so one can read o? results for p odd from p even. The literature is vast for Schr¨ odinger operators with periodic potential called Hill’s equation after Hill [53]. The theory up to the 1950’s is summarized in Magnus-Winkler [83] whose key tool is the discriminant; see also Reed-Simon [115]. There was an explosion of ideas following the KdV revolution, including spectrally invariant ?ows and abelian functions on hyperelliptic Riemann surfaces. Key papers include McKean-van Moerbeke [85], Dubrovin et al. [29], and Trubowitz [144]. Their ideas have been discussed for OPRL; see especially Toda [142], van Moerbeke [145], and Flaschka-McLaughlin [32]. For OPUC, the study of measures associated with (5.1) goes back to Geronimus [37] with a fundamental series of papers by PeherstorferSteinbauer [103, 104, 107, 108, 109, 110, 105, 106] and considerable literature on the case p = 1 (i.e., constant α); see, for example, [40,

OPUC: NEW RESULTS

25

41, 45, 46, 51, 50, 68, 69]. The aforementioned literature on OPUC used little from the the work on Hill’s equation; work that does make a partial link is Geronimo-Johnson [35], which discussed almost periodic Verblunsky coe?cients using abelian functions. Simultaneous with our work reported here, Geronimo-Gesztesy-Holden [33] have discussed this further, including work on isospectral ?ows. Besides the work reported here, Nenciu-Simon [93] have found a symplectic structure on Dp for which the coe?cients of the discriminant Poisson commute (this is discussed in [125, Section 11.11]. 5.1. Discriminant and Floquet Theory. For Schr¨ odinger operators, it is known that the discriminant is just the trace of the transfer matrix. Since the transfer matrix has determinant one in this case, the eigenvalues obey x2 ? Tr(T )x + 1 = 0, which is the starting point for Floquet theory. For OPUC, the transfer matrix, Tp (z ), of (4.5) has det(Tp (z )) = z p , so it is natural to de?ne the discriminant by ?(z ) = z ?p/2 Tr(Tp (z )) (5.2)

which explains why we take p even. Because for z = eiθ , A(α, z ) ∈ U(1, 1) (see [125, Section 10.4] for a discussion of U(1, 1)), ?(z ) is real on ? D, so ?(1/z ?) = ?(z ) Here are the basic properties of ?: Theorem 5.1. (a) All solutions of ?(z ) ? w = 0 with w ∈ (?2, 2) are simple zeros and lie in ? D (so are p in number). (b) {z | ?(z ) ∈ (?2, 2)} is p disjoint intervals on ? D whose closures B1 , . . . , Bp can overlap at most in single points. The complements where |?(z )| > 2 and z ∈ ? D are “gaps,” at most p in number. (c) On ∪Bj , d? is purely a.c. (i.e., in terms of (1.1), ?s (∪p j =1 Bj ) = 0 p and w(θ) > 0 for a.e. θ ∈ ∪j =1 Bj ). (d) ? (? D\ ∪p j =1 Bj ) consists of pure points only with at most one pure point per gap. (e) For all z ∈ C\{0}, the Lyapunov exponent limn→∞ Tn (z ) 1/n exists and obeys γ (z ) = ?(z ) 1 1 log(z ) + log + 2 p 2 ?2 ?1 4 (5.4) (5.3)

where the branch of square root is taken that maximizes the log.

26

B. SIMON

(f) If B = ∪p j =1 Bj , then the logarithmic capacity of B is given by
p?1

CB =
j =0

(1 ? |αj |2 )1/p

(5.5)

and ?[γ (z ) + log CB ] is the equilibrium potential for B . (g) The density of zeros is the equilibrium measure for B and given in terms of ? by dν (θ) = V (θ) dθ 2π (5.6)

where V (θ) = 0 on ? D\ ∪p j =1 Bj , and on Bj is given by 1 p |? (eiθ )| 4 ? ?2 (eiθ )

V (θ) =

(5.7)

where ? (eiθ ) = (h) ν (Bj ) = 1/p

? ?θ

?(eiθ ).

For proofs, see [125, Section 11.1]. The proofs are similar to those for Schr¨ odinger operators. That the density of zeros is an equilibrium measure has been emphasized by Sa?, Stahl, and Totik [116, 131]. While not expressed as the trace of a transfer matrix, ? is related to the (monic) Tchebychev polynomial, T , of Peherstorfer-Steinbauer [108] by ?(z ) = z ?p/2 CB
?1/2

T (z )

and some of the results in Theorem 5.1 are in their papers. One can also relate ? to periodized CMV matrices, an OPUC version of Floquet theory. As discussed in Section 3.7, Ep (β ) is de?ned by restricting E to sequences obeying un+p = βun for all n. Ep can be written as a p × p matrix with an LM factorization. With Θ given by

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(3.4), Ep (β ) = Lp Mp (β ) ? ? ? Mp (β ) = ? ? ?

?αp?1 Θ1 .. . Θp?3 ρp?1 β

ρp β?1

? ? ? ? ? ? (5.8)

α ? p?1 ?

? ? ? ? Lp = ? ? ? ? Then:

Θ0

..

. .. . .. . Θp?2

? ? ? ? ? ? ?

Theorem 5.2. (a) The following holds:
p?1

det(z ? Ep (β )) =
j =0

(1 ? |αj |2 )1/2p z p/2 [?(z ) ? β ? β ?1 ]

(5.9)

(b) E is a direct integral of Ep (β ). 5.2. Generic Potentials. The following seems to be new; it is an analog of a result [119] for Schr¨ odinger operators. p Notice that for any {αj }j =0 ∈ Dp , one can de?ne a discriminant ?1 ?(z, {αj }p j =0 ) for the period p Verblunsky coe?cients that agree with ?1 {αj }p j =0 for j = 0, . . . , p ? 1.
p Theorem 5.3. The set of {αj }p j =0 ∈ D for which ?(z ) has all gaps open is a dense open set.

[125] has two proofs of this theorem: one in Section 11.10 uses Sard’s theorem and one is perturbation theoretic calculation that if (0) ?1 (0) (0) ?1 |Tr(z, {αj }p + (eiη ? 1)δjk αk }p j =0 )| = 2, then |Tr(z, {αj j =0 )| = 2 (0) 2 (0) 3 2 + 2η (ρk ) |αk | + O(η ). 5.3. Borg’s Theorems. In [12], Borg proved several theorems about the implication of closed gaps. Further developments of Borg’s results for Schr¨ odinger equations or for OPRL are in Hochstadt [54, 55, 56, 57, 58, 59, 60, 61], Clark et al. [16], Trubowitz [144], and Flaschka [31]. In [125, Section 11.14], we prove the following analogs of these results:

28

B. SIMON

Theorem 5.4. If {αj }∞ j =0 is a periodic sequence of Verblunsky coe?cients so supp(d?) = ? D (i.e., all gaps are closed), then αj ≡ 0. [125] has three proofs of this: one uses an analog of a theorem of Deift-Simon [22] that d?/dθ ≥ 1/2π on the essential support of the a.c. spectrum of any ergodic system, one tracks zeros of the Wall polynomials, and one uses the analog of Tchebychev’s theorem for the circle that any monic Laurent polynomial real on ? D has maxz∈? D |L(z )| ≥ 2. Theorem 5.5. If p is even and {αj }∞ j =0 has period 2p, then if all gaps with ?(z ) = ?2 are closed, we have αj +p = αj , and if all gaps with ?(z ) = 2 are closed, then αj +p = ?αj . Theorem 5.6. Let p be even and suppose for some k that αkp+j = αj kp ?1 for all j . Suppose for some labelling of {wj }j =0 of the zeros of the derivative ? ?/?θ labelled counterclockwise, we have |?(wj )| = 2 if j ≡ 0 mod k . Then αp+j = ωαj where ω is a k -th root of unity. The proof of these last two theorems depends on the study of the Carath? eodory function for periodic Verblunsky coe?cients as meromorphic functions on a suitable hyperelliptic Riemann surface. 5.4. Green’s Function Bounds. In [125, Section 10.14], we develop the analog of the Combes-Thomas [17] method for OPUC and prove, for points in ? D\supp(d?), the Green’s function (resolvent matrix elements of (C ? z )?1 with C the CMV matrix) decays exponentially in |n ? m|. The rate of decay in these estimates goes to zero at a rate faster than expected in nice cases. For periodic Verblunsky coe?cients, one expects behavior similar to the free case for OPRL or Schr¨ odinger operators — and that is what we discuss here. An energy z0 ∈ ? D at the edge of a band is called a resonance if supn |?n (z0 )| < ∞. For the family of (0) measures, d?λ , with Verblunsky coe?cients αn = λαn and a given z0 , there is exactly one λ for which z0 is a resonance (for the other values, ?n (z0 , d?λ ) grows linearly in n). Here is the bound we prove in [125, Section 11.12]: Theorem 5.7. Let {αn }∞ n=0 be a periodic family of Verblunsky coe?cients. Suppose G = {z = eiθ | θ0 < θ < θ1 } is an open gap and eiθ0 is not a resonance. Let Gnm (z ) = δn , (C (α) ? z )?1 δm Then for z = eiθ with z ∈ G and |θ ? θ0 | < |θ ? θ1 |, we have sup |Gnm (z )| ≤ C1 |z ? eiθ0 |?1/2
n,m

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such z

sup |Gnm (z )| ≤ C2 (n + 1)1/2 (m + 1)1/2

and similarly for z approaching eiθ1 . The proof depends on bounds on polynomials in the bands of some independent interest. Theorem 5.8. Let {αn }∞ n=0 be a sequence of periodic Verblunsky coef?cients, and let B int be the union of the interior of the bands. Let E1 be the set of band edges by open gaps and E2 the set of band edges by closed gaps. De?ne d(z ) = min(dist(z, E1 ), dist(z, E2 )2 ) Then (1) (2) sup |?n (z )| ≤ C1 d(z )?1/2
n z ∈B int

sup |?n (z )| ≤ C2 n

where C1 and C2 are {αn }∞ n=0 dependent constants. Remark. One can, with an extra argument, show d(z ) can be replaced by dist(z, E1 ) which di?ers from d(z ) only when there is a closed gap. That is, there is no singularity in supn |?n (z )| at band edges next to closed gaps. 5.5. Isospectral Results. In [125, Chapter 11], we prove the following theorem:
?1 p?1 p Theorem 5.9. Let {αj }p j =0 be a sequence in D so ?(z, {αj }j =0 ) has ?1 p?1 p?1 p k open gaps. Then {{βj }p j =0 ∈ D | ?(z, {βj }j =0 ) = ?(z, {αj }j =0 )} is a k -dimensional torus.

This result for OPUC seems to be new, although its analog for ?nitegap Jacobi matrices and Schr¨ odinger operators (see, e.g., [85, 29, 145]) is well known and it is related to results on almost periodic OPUC by Geronimo-Johnson [35]. There is one important di?erence between OPUC and the Jacobi/Schr¨ odinger case. In the later, the in?nite gap doesn’t count in the calculation of dimension of torus, so the torus has a dimension equal to the genus of the Riemann surface for the m-function. In the OPUC case, all gaps count and the torus has dimension one more than the genus. The torus can be de?ned explicitly in terms of natural additional ?1 data associated to {αj }p j =0 . One way to de?ne the data is to analytically continue the Carath? eodory function, F , for the periodic sequence.

30

B. SIMON

One cuts C on the “combined bands,” that is, connected components of {eiθ | |?(√ eiθ )| ≤ 2}, and forms the two-sheeted Riemann surface associated to ?2 ? 4. On this surface, F is meromorphic with exactly one pole on each “extended gap.” By extended gap, we mean the closure of the two images of a gap on each of two sheets of the Riemann surface. The ends of the gap are branch points and join the two images into a circle. The p points, one on each gap, are thus p-dimensional torus, and the re?ned version of Theorem 5.9 is that there is exactly one Carath? eodory function associated to a period p set of Verblunsky coe?cients with speci?ed poles. Alternately, the points in the gaps are solutions of Φp (z ) ? Φ? p (z ) = 0 with sheets determined by whether the points are pure points of the associated measure or not. [125] has two proofs of Theorem 5.9: one using the Abel map on the above referenced Riemann surface and one using Sard’s theorem. 5.6. Perturbation Conjectures. [124, 125] have numerous conjectures and open problems. We want to end this section with a discussion of conjectures that describe perturbations of periodic Verblunsky coe?cients. We discuss the Weyl-type conjecture in detail. As a model, consider Theorem 3.4 when αn ≡ a = 0. For ess supp(dν ) to be Γa,1 , the essential support for αn ≡ a, it su?ces that |αn | → a and αn+1 /αn → 1. This suggests Conjecture 5.10. Fix a period p set of Verblunsky coe?cients with discriminant ?. Let M be the set of period p (semi-in?nite) sequences with discriminant ? and let S ? ? D be their common essential support. Suppose
∞ j →∞ α∈M

lim inf

e?n |βj +n ? αn | = 0
n=1

Then if ν is the measure with Verblunsky coe?cients β , then ess supp(dν ) = S . Thus, limit results only hold in the sense of approach to the isospectral manifold. There are also conjectures in [125] for extensions of Szeg? o’s and Rakmanov’s theorems in this context. 6. Spectral Theory Examples [125, Chapter 12] is devoted to analysis of speci?c classes of Verblunsky coe?cients, mainly ?nding analogs of known results for Schr¨ odinger or discrete Schr¨ odinger equations. Most of these are reasonably

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straightforward, but there are often some extra tricks needed and the results are of interest. 6.1. Sparse and Decaying Random Verblunsky Coe?cients. In [72], Kiselev, Last, and Simon presented a thorough analysis of continuum and discrete Schr¨ odinger operators with sparse or decaying random potentials, subjects with earlier work by Pearson [102], Simon [121], Delyon [23, 24], and Kotani-Ushiroya [74]. In [125, Sections 12.3 and 12.7], I have found analogs of these results for OPUC: Theorem 6.1. Let d? have the form (1.1). Let {n }∞ =1 be a monotone n +1 sequence of positive integers with lim inf →∞ n > 1 and αj (d?) = 0 and


if j ∈ / {n }

(6.1) (6.2)

|αj (d?)|2 < ∞
j =0 dθ p Then ?s = 0, supp(d?) = ? D, and w, w?1 ∈ ∩∞ p=1 L (? D, 2π ).

This result was recently independently obtained by Golinskii [48]: Theorem 6.2. Let {n }∞ =1 be a monotone sequence of positive integers n +1 with lim n = ∞ so that (6.1) holds. Suppose limj →∞ |αj (d?)| = 0 and (6.2) fails. Then d? is purely singular continuous. Theorem 6.3. Let {αj (ω )}∞ j =0 be a family of independent random variables with values in D with E(αj (ω )) = 0 and


(6.3) (6.4)

E(|αj (ω )|2 ) < ∞
j =0

Let d?ω be the measure with αj (d?ω ) = αj (ω ). Then for a.e. ω , d?ω has the form (1.1) with d?ω,s and w(θ) > 0 for a.e. θ. This result is not new; it is a result of Teplyaev, with earlier results of Nikishin [99] (see Teplyaev [138, 139, 140, 141]). We state it for comparison with the next two theorems. The theorems assume (6.3) and also sup |αj (ω )| < 1
ω,j

sup |αj (ω )| → 0
ω

as j → ∞

(6.5) (6.6)

E(αj (ω )2 ) = 0 E(|αj (ω )|2 )1/2 = Γj ?γ if j > J0

(6.7)

32

B. SIMON

Theorem 6.4. If {αj (ω )}∞ j =0 is a family of independent random vari1 ables so (6.3), (6.5), (6.6), and (6.7) hold and Γ > 0, γ < 2 , then for a.e. pairs ω and λ ∈ ? D, d?λ,ω , the measure with αj (d?λ,ω ) = λαj (ω ), is pure point with support equal to ? D (i.e., dense mass points). Theorem 6.5. If {αj (ω )}∞ j =0 is a family of independent random vari1 ables so (6.3), (6.5), (6.6), and (6.7) hold for Γ > 0, γ = 2 , and sup n1/2 |αn (ω )| < ∞
n,ω

(6.8)

Then (i) If Γ2 > 1, then for a.e. pairs λ ∈ ? D, ω ∈ ?, d?λ,ω has dense pure point spectrum. (ii) If Γ2 ≤ 1, then for a.e. pairs λ ∈ ? D, ω ∈ ?, d?λ,ω has purely singular continuous spectrum of exact Hausdor? dimension 1 ? Γ2 in that d?λ,ω is supported on a set of dimension 1 ? Γ2 and gives zero weight to any set S with dim(S ) < 1 ? Γ2 . For the last two theorems, a model to think of is to let {βn }∞ n=0 be identically distributed random variables on {z | |z | ≤ r} for some r < 1 with a rotationally invariant distribution and to let αn = Γ1/2 E(|β1 |2 )?1/2 max(n, 1)?γ βn . The proofs of these results exploit Pr¨ ufer variables, which for OPUC go back to Nikishin [98] and Nevai [94]. 6.2. Fibonacci Subshifts. For discrete Schr¨ odinger operators, there is an extensive literature [4, 73, 100, 133, 9, 13, 19, 20, 18, 21, 79, 80] on subshifts (see [125, 112, 82] for a de?nition of subshifts). In [125, Section 12.8], we have analyzed the OPUC analog of the most heavily studied of these subshifts, de?ned as follows: Pick α, β ∈ D. Let F1 = α, F2 = αβ , and Fn+1 = Fn Fn?1 for n = 2, 3, . . . . Fn+1 is a sequence which starts with Fn and so there is a limit F = α, β, α, α, β, α, β, α, α, β, α, α, . . . . We write F (α, β ) when we want to vary α and β . Theorem 6.6. The essential support of the measure ? with α (d?) = F (α, β ) is a closed perfect set of Lebesgue measure zero for any α = β . For ?xed α0 , β0 and a.e. λ ∈ ? D, the measure with α (d?) = F (λα0 , λβ0 ) is a pure point measure, with each pure point isolated and dθ -measure zero. the limit points of the pure points a perfect set of 2 π The proof follows that for Schr¨ odinger operators with a few additional tricks needed.

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6.3. Dense Embedded Point Spectrum. Naboko [87, 88, 89, 90, d2 91] and Simon [122] constructed Schr¨ odinger operators ? dx 2 + V (x) with V (x) decaying only slightly slower than |x|?1 so there is dense embedded point spectrum. Naboko’s method extends to OPUC. Theorem 6.7. Let g (n) be an arbitrary function with 0 < g (n) ≤ g (n + 1) and g (n) → ∞ as n → ∞. Let {ωj }∞ j =0 be an arbitrary subset of ? D which are multiplicatively rationally independent, that is, for no n1 , n2 , . . . , nk ∈ Z other than (0, 0, . . . , 0), is it true that k ?1 nj = 1. Then there exists a sequence {αj }∞ j =0 of Verblunj =1 (ωj ω0 ) sky coe?cients with g (n) |α n | ≤ n for all n so that the measure d? with αj (d?) = αj has pure points at each ωj . Remark. If g (n) ≤ n1/2?ε , then |αn | ∈ 2 so, by Szeg? o’s theorem, d? has the form (1.1) with w(θ) > 0 for a.e. θ, that is, the point masses are embedded in a.c. spectrum. 6.4. High Barriers. Jitomirskaya-Last [65] analyzed sparse high barriers to get discrete Schr¨ odinger operators with fractional-dimensional spectrum. Their methods can be applied to OPUC. Let 0 < a < 1 and L = 2n
n

(6.9) (6.10) j = Ln otherwise (6.11)
?(1?a)/2a

?1/2 αj = (1 ? ρ2 j)

ρj =

Ln 0

Theorem 6.8. Let αj be given by (6.10)/ (6.11) and let d?λ be the Aleksandrov measures with αj (d?λ ) = λαj . Then for Lebesgue a.e. λ, d?λ has exact dimension a in the sense that d?λ is supported on a set of Hausdor? dimension a and gives zero weight to any set B of Hausdor? dimension strictly less than a.

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