Infinite gap Jacobi matrices - PDF Free Download

Infinite gap Jacobi matrices Jacob Stordal Christiansen Centre for Mathematical Sciences, Lund University Spectral Theory of Orthogonal Polynomials Master class by Barry Simon Centre for Quantum Geometry of Moduli Spaces Århus University

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Outline Parreau Widom sets a large class of compact subsets of the real line R 2 / 29

Outline Parreau Widom sets a large class of compact subsets of the real line R Szegő class theory including asymptotics of the orthogonal polynomials 2 / 29

Literature M. Sodin and P. Yuditskii. Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7 (1997) 387-435 F. Peherstorfer and P. Yuditskii. Asymptotic behavior of polynomials orthonormal on a homogeneous set. J. Anal. Math. 89 (2003) 113 154 J. S. Christiansen. Szegő s theorem on Parreau Widom sets. Adv. Math. 229 (2012) 1180 1204 C. Remling. The absolutely continuous spectrum of Jacobi matrices. Ann. Math. 174 (2011) 125 171 P. Yuditskii. On the Direct Cauchy Theorem in Widom domains: Positive and negative examples. Comput. Methods Funct. Theory 11, 395 414 (2011) J. S. Christiansen. Dynamics in the Szegő class and polynomial asymptotics. 3 / 29

Parreau Widom sets Let E R be a compact set of positive logarithmic capacity and denote by g the Green s function for Ω = C E with pole at. 4 / 29

Parreau Widom sets Let E R be a compact set of positive logarithmic capacity and denote by g the Green s function for Ω = C E with pole at. Assume that each point of E is a regular point for Ω and let dµ E be the equilibrium measure of E. Recall that and write g(x) = log t x dµ E (t) log(cap(e)) E = [α, β] j (α j, β j ). While g vanishes on E, it is concave on (α j, β j ) for each j. So there is precisely one critical point c j per gap. 4 / 29

Comb-like domains E 6 / 29

Comb-like domains E 7 / 29

Comb-like domains E dµ E is absolutely continuous iff sup j { g(c j ) v j v } < for a.e. v (0, π). 7 / 29

Comb-like domains E dµ E is absolutely continuous iff sup j { g(c j ) v j v } < for a.e. v (0, π). This is always the case when j g(c j ) < (i.e., E is PW). 7 / 29

Homogeneous sets Any finite gap set is PW, but the notion goes way beyond. 8 / 29

Homogeneous sets Any finite gap set is PW, but the notion goes way beyond. It includes all compact sets E that are homogeneous in the sense of Carleson. By definition, this means there is an ε > 0 so that (t δ, t + δ) E ε for all t E and all δ < diam(e). δ This geometric condition was introduced to avoid cases where certain parts of E are very thin compared to Lebesgue measure. Example Remove the middle 1/4 from [0, 1] and continue removing subintervals of length 1/4 n from the middle of each of the 2 n 1 remaining intervals. Let E be the set of what is left in [0, 1] a fat Cantor set of E = 1/2. One can show that (t δ, t + δ) E δ/4 for all t E and all δ < 1. 8 / 29

Gábor Szegő...on sabbatical Lehrsätze aus der Analysis, vols. I and II (writt with his mentor and frien George (György) Póly Springer-Verlag, Berlin, 192 which was used by gene tions of mathematics studen (and their professors). Qu ing Pólya: It was a wonder time; we worked with enth siasm and concentration. W had similar backgrounds. W were both influenced, like other young Hungarian ma ematicians of that time, Leopold (Lipót) Fejér. We we both readers of the same we directed Hungarian Mat ematical Journal for hi school students that stress problem solving. We were terested in the same kind questions, in the same topi but one of us knew mo about one topic, and the oth more about some other top It was a fine collaboration. The book Aufgab und Lehrsätze aus der Analysis, the result of o 9 / 29

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Szegő s theorem on PW sets Let E R be a Parreau Widom set and let J = {a n, b n } n=1 be a Jacobi matrix with spectral measure dρ = f (t)dt + dρ s. 12 / 29

Szegő s theorem on PW sets Let E R be a Parreau Widom set and let J = {a n, b n } n=1 be a Jacobi matrix with spectral measure dρ = f (t)dt + dρ s. Assume that σ ess (J) = E and denote by {x k } the eigenvalues of J outside E, if any. [Adv. Math. 2012] On condition that k g(x k ) <, we have E log f (t)dµ E (t) > lim sup n In the affirmative, 0 < lim inf n a 1 a n Cap(E) n > 0. a 1 a n a 1 a n lim sup Cap(E) n n Cap(E) n <. 12 / 29

The Szegő class Inspired by Szegő s theorem, we introduce the class Sz(E). 13 / 29

The Szegő class Inspired by Szegő s theorem, we introduce the class Sz(E). Defn. A Jacobi matrix with spectral measure dρ = f (t)dt + dρ s is said to belong to the Szegő class for E if the essential support of dρ is equal to E, the absolutely continuous part of dρ obeys the Szegő condition E log f (t)dµ E (t) >, the isolated mass points x k of dρ outside E satisfy the condition k g(x k ) <. 13 / 29

Reflectionless operators Given a PW set E, we denote by T E the set of all two-sided Jacobi matrices J = {a n, b n} n= that have spectrum equal to E and are reflectionless on E, that is, Re δ n, (J (t + i0)) 1 δ n = 0 for a.e. t E and all n. Equivalently, J = (a n) 2 m + n (t + i0) = b n 1 a n 1 a n 1 b n a n a n b n+1 a n+1 a n+1 b n+2 1 m n (t + i0) for a.e. t E and all n, where m + n is the m-function for J + n = {a n+k, b n+k } k=1 and m n the m-function for J n = {a n k, b n+1 k } k=1. 14 / 29

Remling s theorem By compactness, any bounded J = {a n, b n } n=1 has accumulation points when the coefficients are shifted to the left. Such two-sided limit points are also called right limits of J. Let E R be a compact set and assume that E > 0. If σ ess (J) = E and the spectral measure dρ = f (t)dt + dρ s of J obeys f (t) > 0 for a.e. x E, then any right limit of J belongs to T E. [Ann. of Math. 2011] 15 / 29

The collection of divisors Recall that E = [α, β] j (α j, β j ). 16 / 29

Recall that The collection of divisors E = [α, β] j (α j, β j ). As described below, there is a natural way to introduce a torus of dimension equal to the number of gaps in E. 16 / 29

Recall that The collection of divisors E = [α, β] j (α j, β j ). As described below, there is a natural way to introduce a torus of dimension equal to the number of gaps in E. The set D E of divisors consists of all formal sums D = j (y j, ±), y j [α j, β j ], where (y j, +) and (y j, ) are identified when y j is equal to α j or β j. 16 / 29

The collection of divisors Recall that E = [α, β] j (α j, β j ). university of copenhagen department of mathematical sciences As described below, there is a natural way to introduce a torus of dimension equal to the number of gaps in E. The set D E of divisors consists of all formal sums D = j (y j, ±), y j [α j, β j ], where (y j, +) and (y j, ) are identified when y j is equal to α j or β j. α α j β j α k β k β 16 / 29

A map T E D E When J T E, we know that G(x) = δ 0, (J x) 1 δ 0 is analytic on C E and has purely imaginary boundary values a.e. on E. 17 / 29

A map T E D E When J T E, we know that G(x) = δ 0, (J x) 1 δ 0 is analytic on C E and has purely imaginary boundary values a.e. on E. Such Nevanlinna Pick functions admit a representation of the form G(x) = where y j [α j, β j ] for each j. Using the relation 1 x y j (x α)(x β) j (x αj )(x β j ), (a 0) 2 m + (x) 1/m (x) = 1/G(x), it follows that every y j (α j, β j ) is a pole of either m + or 1/m. 17 / 29

Fuchsian groups & charaters The universal covering map x D Ω (= C E) is onto but only locally one-to-one. 18 / 29

Fuchsian groups & charaters The universal covering map x D Ω (= C E) is onto but only locally one-to-one. Associated with x is a Fuchsian group Γ E of Möbius transformations on D such that x(z) = x(w) γ Γ E z = γ(w). Γ E is isomorphic to the fundamental group π 1 (Ω) and hence a free group on as many generators as the number of gaps in E. We denote by ΓE the multiplicative group of unimodular characters on Γ E and equip it with the topology dual to the discrete one. 18 / 29

The Abel map The Abel map D E ΓE, usually defined via Abelian integrals, is well-defined and continuous when E is a PW set. 19 / 29

The Abel map The Abel map D E ΓE, usually defined via Abelian integrals, is well-defined and continuous when E is a PW set. A key result of Sodin Yuditskii states that the maps T E D E Γ E are homeomorphisms when the direct Cauchy theorem holds. Here, the topology on T E is induced by operator norm and every point in T E has almost periodic Jacobi parameters. There exist PW sets for which DCT fails, but PW with DCT is still more general than homogeneous. In fact, the Abel map is bijective if and only if DCT holds. 19 / 29

The Abel map The Abel map D E ΓE, usually defined via Abelian integrals, is well-defined and continuous when E is a PW set. A key result of Sodin Yuditskii states that the maps T E D E Γ E are homeomorphisms when the direct Cauchy theorem holds. Here, the topology on T E is induced by operator norm and every point in T E has almost periodic Jacobi parameters. There exist PW sets for which DCT fails, but PW with DCT is still more general than homogeneous. From our point of view, the map T E ΓE is given by the character of the Jost function of J (to be defined shortly). In fact, the Abel map is bijective if and only if DCT holds. 19 / 29

blaschke products To every w D, we associated the blaschke product γ(w) γ(w) z B(z, w) = γ Γ E γ(w) 1 γ(w)z. 20 / 29

blaschke products To every w D, we associated the blaschke product γ(w) γ(w) z B(z, w) = γ Γ E γ(w) 1 γ(w)z. Not only is B(, w) analytic on D with simple zeros at {γ(w)} γ ΓE, it is also character automorphic. This means there is a character χ w in Γ E such that B(γ(z), w) = χ w (γ)b(z, w). In particular, the character of B(z) = B(z, 0) is given by χ 0 (γ j ) = exp{2πi µ E ([β j, β])} for a suitable choice {γ j } of generators of Γ E. 20 / 29

The direct Cauchy theorem As before, let {c j } denote the critical points of g (the Green s function for Ω with pole at ). 21 / 29

The direct Cauchy theorem As before, let {c j } denote the critical points of g (the Green s function for Ω with pole at ). Pick w j D such that x(w j ) = c j and form the blaschke product c(z) = j B(z, w j ). When E is a PW set, this product converges to a character automorphic function on D with character χ c. Definition The direct Cauchy theorem (DCT) is said to hold if 0 2π ϕ(e iθ ) dθ c(e iθ ) 2π = ϕ(0) c(0) whenever ϕ H 1 (D) is character automorphic and χ ϕ = χ c. 21 / 29

Szegő asymptotics Let E R be a Parreau Widom set and assume that the direct Cauchy theorem holds. 22 / 29

Szegő asymptotics Let E R be a Parreau Widom set and assume that the direct Cauchy theorem holds. Suppose that J = {a n, b n } n=1 belongs to the Szegő class for E. Then there is a unique J = {a n, b n} n= in T E such that a n a n + b n b n 0. Consequently, a n and b n are asymptotically almost periodic. Moreover, if dµ is the spectral measure of J restricted to l 2 (N), then P n (x, dµ) / P n (x, dµ ) has a limit for all x C R. 22 / 29

The Jost function Let us introduce the key player for polynomial asymptotics. 23 / 29

The Jost function Let us introduce the key player for polynomial asymptotics. Definition Given J Sz(E), we define the Jost function by u(z; J) = k B(z, p k ) exp { 1 2 2π 0 e iθ + z e iθ z log f (x(eiθ )) dθ 2π }, where the p k s are chosen in such a way that x(p k ) = x k, the eigenvalues of J in R E. The Jost function is analytic on D and one can show that it is also character automorphic, that is, χ Γ E u(γ(z); J) = χ(γ) u(z; J) for all γ Γ E. As for notation, we denote by χ(j) the character of J. 23 / 29

Elements of the proof The proof of the first part relies on three important results: 24 / 29

Elements of the proof The proof of the first part relies on three important results: 1 the Denisov Rakhmanov Remling theorem any right limit of J belongs to T E 24 / 29

Elements of the proof The proof of the first part relies on three important results: 1 the Denisov Rakhmanov Remling theorem any right limit of J belongs to T E 2 the Jost isomorphism of Sodin and Yuditskii the map T E J χ(j ) Γ E is a homeomorphism 24 / 29

Proof, first part Let J be the unique element in T E for which χ(j ) = χ(j). 25 / 29

Proof, first part Let J be the unique element in T E for which χ(j ) = χ(j). For contradiction, suppose that a n a n + b n b n / 0. 25 / 29

Proof, first part Let J be the unique element in T E for which χ(j ) = χ(j). For contradiction, suppose that a n a n + b n b n / 0. Then there is a subsequence {m l } so that J and J have different right limits, say K K. According to Remling s theorem, both K and K lie in T E. Moreover, by Jost asymptotics for right limits, we have χ(j ml ) χ(k) and χ(j m l ) χ(k ). 25 / 29

Proof, second part The Jost solution defined by u n (z; J) = a 1 n B(z) n u(z; J n ) satisfies the same three-term recurrence relation as P n 1. 26 / 29

Proof, second part The Jost solution defined by u n (z; J) = a 1 n B(z) n u(z; J n ) satisfies the same three-term recurrence relation as P n 1. So we can write the diagonal Green s function as G nn (x(z), dµ) = P n 1 (x(z), dµ)u n (z; J)/Wr(z), where Wr(z) is the Wronskian which is equal to u(z; J). Now, P n 1 (x(z), dµ) P n 1 (x(z), dµ ) = G nn(x(z), dµ) u n (z; J ) u(z; J) G nn (x(z), dµ ) u n (z; J) u(z; J ). 26 / 29

The proof shows that Polynomial asymptotics P n (x(z), dµ) u(z; J) P n (x(z), dµ ) u(z; J ) locally uniformly on F int, a fundamental domain for Γ E. 27 / 29

The proof shows that Polynomial asymptotics P n (x(z), dµ) u(z; J) P n (x(z), dµ ) u(z; J ) locally uniformly on F int, a fundamental domain for Γ E. The orthonormal polynomials associated with points in T E behave like a nb(z) n P n (x(z), dµ ) u(z; dµ )u(z; dµ,r n 1 ), where J,r is the matrix given by a,r n = a n 1, b,r n = b n for n Z. 27 / 29

Open problems Let E R be a Parreau Widom set and assume that the direct Cauchy theorem holds. 28 / 29

Open problems Let E R be a Parreau Widom set and assume that the direct Cauchy theorem holds. Suppose J = {a n, b n} n= is a matrix in T E and J = {a n, b n } n=1 is an arbitrary Jacobi matrix. 28 / 29

Open problems Let E R be a Parreau Widom set and assume that the direct Cauchy theorem holds. Suppose J = {a n, b n} n= is a matrix in T E and J = {a n, b n } n=1 is an arbitrary Jacobi matrix. Conjecture: If a n a n + b n b n <, then J belongs to the Szegő class for E. Conjecture: If J lies in the Szegő class for E and χ(j) = χ(j ), then (a n a n) 2 + (b n b n) 2 <. Open question: If (a n a n) 2 + (b n b n) 2 < for some J, what can we say about J? 28 / 29

Thank you very much for your attention! 29 / 29