Orthogona l Polynomial s on the Uni t Circl e Par t 2: Spectra l Theor y This page intentionally left blank http://dx.doi.org/10.1090/coll/054.2 America n Mathematica l Societ y Colloquiu m Publication s Volum e 54 , Par t 2 Orthogona l Polynomial s on the Uni t Circl e Par t 2: Spectra l Theor y Barr y Simo n America n Mathematica l Societ y Providence , Rhod e Islan d Editorial Boar d Susan J . Friedlander , Chai r Yuri Mani n Peter Sarna k 2000 Mathematics Subject Classification. Primar y 42C05 , 05E35 , 34L99 ; Secondary 47B35 , 30C85 , 30D55 , 42A10 . For additiona l Informatio n an d Update s o n thi s book , visi t www.ams.org/bookpages/coll-54 Library o f Congres s Cataloging-in-Publicatio n Dat a Simon, Barry , 1946 - Orthogonal polynomial s o n the uni t circl e / Barr y Simon . p. cm . - (America n Mathematica l Societ y colloquiu m publications , ISS N 0065-925 8 ; v. 54 ) Contents: pt . 1 . Classica l theor y Includes bibliographica l reference s an d index . 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Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 0 0 9 To my grandchildren an d thei r parent s This page intentionally left blank Contents Preface t o Part 1 x i Notation xvi i Chapter 1 Th e Basic s 1 1.1 Introductio n 1 1.2 Orthogona l Polynomial s o n the Real Line 1 1 1.3 Caratheodor y an d Schu r Function s 2 5 1.4 A n Introduction t o Operato r an d Spectra l Theor y 4 0 1.5 Verblunsk y Coefficient s an d the Szeg ö Recurrence 5 5 1.6 Example s o f OPUC 7 1 1.7 Zero s and the Firs t Proo f o f Verblunsky's Theore m 9 0 Chapter 2 Szegö' s Theorem 10 9 2.1 Toeplit z Determinant s an d Verblunsk y Coefficient s 10 9 2.2 Extrema l Properties , the ChristofFe l Functions , an d th e Christoffel-Darboux Formul a 11 7 2.3 Entrop y Semicontinuit y an d the First Proo f o f Szegö' s Theorem 13 6 2.4 Th e Szeg ö Function 14 3 2.5 Szegö' s Theorem Usin g the Poisso n Kerne l 15 1 2.6 Khrushchev' s Proo f o f Szegö' s Theorem 15 6 2.7 Consequence s o f Szegö's Theorem 15 9 2.8 A Higher-Order Szeg ö Theorem 17 2 2.9 Th e Relativ e Szeg ö Function 17 8 2.10 Totik' s Worksho p 18 4 2.11 Ries z Products an d Khrushchev' s Worksho p 18 9 2.12 Th e Workshop o f Denisov an d Kupi n 19 7 2.13 Matrix-Value d Measure s 20 6 Chapter 3 Tool s fo r Geronimus ' Theore m 21 7 3.1 Verblunsky' s Viewpoint : Proof s o f Verblunsky's an d Geronimus' Theorem s 21 7 3.2 Secon d Kin d Polynomial s 22 2 3.3 K W Pair s 23 9 3.4 Coefficien t Strippin g an d Associate d Polynomial s 24 5 Chapter 4 Matri x Representation s 25 1 4.1 Th e GGT Representatio n 25 1 4.2 Th e CM V Representation 26 2 4.3 Spectra l Consequence s o f the CM V Representation 27 4 4.4 Th e Resolven t o f the CM V Matrix 28 7 viii CONTENT S 4.5 Ran k Tw o Perturbations an d Decouplin g o f CMV Matrice s 29 3 Chapter 5 Baxter' s Theore m 30 1 5.1 Wiener-Hop f Factorizatio n an d the Inverse s o f Finite Toeplit z Matrice s 30 1 5.2 Baxter' s Proo f 31 3 Chapter 6 Th e Stron g Szeg ö Theorem 31 9 6.1 Th e Ibragimo v an d Golinskii-Ibragimo v Theorem s 31 9 6.2 Th e Borodin-Okounko v Formul a 33 3 6.3 Representation s o f U(n) an d th e Bump-Diaconi s Proo f 34 6 6.4 Toeplit z Determinant s a s the Statistica l Mechanic s o f Coulomb Gase s an d Johansson' s Proo f 35 2 6.5 Th e Combinatoria l Approac h an d Kac' s Proo f 36 8 6.6 A Secon d Loo k a t Ibragimov' s Theore m 37 6 Chapter 7 Verblunsk y Coefficient s Wit h Rapi d Deca y 38 1 7.1 Th e Rate o f Exponential Deca y an d a Theorem o f Nevai-Totik 38 1 7.2 Detaile d Asymptotic s o f the Verblunsk y Coefficient s 38 7 Chapter 8 Th e Densit y o f Zero s 39 1 8.1 Th e Densit y o f Zero s Measure vi a Potential Theor y 39 1 8.2 Th e Densit y o f Zero s Measure vi a the CM V Matrix 40 3 8.3 Rotatio n Number s 41 0 8.4 A Gallery o f Zero s 41 2 Bibliography 42 5 Author Inde x 45 7 Subject Inde x 46 3 Preface t o Part 2 x i Notation xii i Chapter 9 Rakhmanov' s Theore m an d Relate d Issue s 46 7 9.1 Rakhmanov' s Theore m vi a Polynomial Ratio s 46 7 9.2 Khrushchev' s Proo f o f Rakhmanov's Theore m 47 5 9.3 Furthe r Aspect s o f Khrushchev's Theor y 48 5 9.4 Introductio n t o MNT Theor y 49 3 9.5 Rati o Asymptotic s 50 3 9.6 Poincare' s Theore m an d Rati o Asymptotic s 51 2 9.7 Wea k Asymptotic Measure s 52 1 9.8 Rati o Asymptotics fo r Varyin g Measure s 53 0 9.9 Rakhmanov' s Theore m o n an Ar e 53 5 9.10 Wea k Limits an d Relativ e Szeg ö Asymptotics 53 8 Chapter 1 0 Technique s o f Spectral Analysi s 54 5 10.1 Aronszajn-Donoghu e Theor y 54 5 10.2 Spectra l Averagin g an d th e Simon-Wolf f Criterio n 55 1 10.3 Th e Gordon-de l Rio-Makarov-Simo n Theore m 55 8 10.4 Th e Grou p U(l, 1) 56 4 CONTENTS i x 10.5 Lyapuno v Exponent s an d th e Growt h o f Norms i n U(l, 1 ) 58 1 10.5A Appendix: Subshift s 60 0 10.6 Furstenberg' s Theore m an d Rando m Matri x Product s Prom U(l, 1 ) 60 6 10.7 Th e Transfe r Matri x Approac h to L 1 Verblunsk y Coefficient s 61 7 10.8 Th e Jitomirskaya-Las t Inequalitie s 63 1 10.9 Criteri a fo r A.C . Spectrum 63 9 10.10 Dependenc e o n the Tai l 64 8 10.11 Kotan i Theor y 65 2 10.12 Prüfe r Variable s 66 4 10.13 Modifying th e Measure : Insertin g Eigenvalue s an d Rational Functio n Multiplicatio n 67 3 10.14 Deca y o f CMV Resolvent s an d Eigenfunction s 68 5 10.15 Countin g Eigenvalue s i n Gaps : Th e Birman-Schwinge r Principl e 69 0 10.16 Stochasti c Verblunsk y Coefficient s 70 1 Chapter 1 1 Periodi c Verblunsk y Coefficient s 70 9 11.1 Th e Discriminan t 71 0 11.2 Floque t Theor y 71 9 11.3 Calculatio n o f the Weigh t 72 4 11.4 A n Overvie w o f the Invers e Spectra l Proble m 73 0 11.5 Th e Orthogona l Polynomial s Associate d t o Dirichle t Dat a 74 2 11.6 Wal l Polynomials an d the Determinatio n o f Discriminants 74 8 11.7 Abel' s Theore m an d the Invers e Spectra l Proble m 75 3 11.8 Almos t Periodi c Isospectra l Tor i 78 3 11.9 Quadrati c Irrationalitie s 78 8 11.10 Independenc e o f Spectral Invariants an d Isospectra l Tor i 79 9 11.11 Isospectra l Flow s 80 1 11.12 Bound s o n the Green' s Functio n 80 8 11.13 Genericit y Result s 81 1 11.14 Consequence s o f Many Close d Gap s 81 2 Chapter 1 2 Spectra l Analysi s o f Specifi c Classe s of Verblunsky Coefficient s 81 7 12.1 Perturbation s o f Bounded Variatio n 81 7 12.2 Perturbation s o f Periodic Verblunsk y Coefficient s 82 6 12.3 Naboko' s Workshop : Dens e Point Spectru m i n the Szeg ö Class 82 9 12.4 Generi c Singula r Continuou s Spectru m 83 4 12.5 Spars e Verblunsk y Coefficient s 83 8 12.6 Rando m Verblunsk y Coefficient s 84 5 12.7 Decayin g Rando m Verblunsk y Coefficient s 84 7 12.8 Subshift s 85 5 12.9 Hig h Barriers 86 3 Chapter 1 3 Th e Connectio n t o Jacob i Matrice s 87 1 13.1 Th e Szeg ö Mapping an d Geronimu s Relation s 87 1 13.2 CM V Matrice s an d the Geronimu s Relation s 88 1 13.3 Szegö' s Theorem fo r OPRL : A First Loo k 88 9 13.4 Th e Denisov-Rakhmano v Theore m 89 2 13.5 Th e Damanik-Killi p Theore m 89 6 x CONTENT S 13.6 Th e Geronimo-Cas e Equation s 90 3 13.7 Jacob i Matrice s With Exponentiall y Decayin g Coefficient s 91 2 13.8 Th e P 2 Su m Rule and Application s 92 0 13.9 Szegö' s Theorem fo r OPRL : A Third Loo k 93 7 Appendix A Reader' s Guide : Topic s and Formula e 94 5 A.l What' s Don e Where 94 5 A.