What is spectral theory? Spectral Theory of Orthogonal Polynomials OPs
OPRL basics Favard's Theorem Barry Simon OPUC basics IBM Professor of Mathematics and Theoretical Physics Szeg® recursion and Verblunsky California Institute of Technology coe cients Pasadena, CA, U.S.A. BernsteinSzeg® Approximation
Carmona Simon Formula Lecture 1: Introduction and Overview Spectral Theory of Orthogonal Polynomials
What is spectral theory?
OPs OPRL basics Lecture 1: Introduction and Overview Favard's Theorem Lecture 2: Szegö Theorem for OPUC OPUC basics
Szeg® recursion Lecture 3: Three Kinds of Polynomials Asymptotics, I and Verblunsky coe cients Lecture 4: Three Kinds of Polynomial Asymptotics, II BernsteinSzeg® Approximation
Carmona Simon Formula References
[OPUC] B. Simon, Orthogonal Polynomials on the Unit What is spectral Circle, Part 1: Classical Theory, AMS Colloquium Series theory?
OPs 54.1, American Mathematical Society, Providence, RI,
OPRL basics 2005. Favard's Theorem [OPUC2] B. Simon, Orthogonal Polynomials on the Unit OPUC basics Circle, Part 2: Spectral Theory, AMS Colloquium Series, Szeg® recursion and Verblunsky 54.2, American Mathematical Society, Providence, RI, coe cients 2005. BernsteinSzeg® Approximation [SzThm] B. Simon, Szeg®'s Theorem and Its Descendants: Carmona Simon Formula Spectral Theory for L2 Perturbations of Orthogonal Polynomials, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011. What is spectral theory?
What is spectral theory?
OPs
OPRL basics Spectral theory is the general theory of the relation of the Favard's Theorem fundamental parameters of an object and its spectral OPUC basics characteristics. Szeg® recursion and Verblunsky Spectral characteristics means eigenvalues or scattering coe cients data or, more generally, spectral measures BernsteinSzeg® Approximation
Carmona Simon Formula What is spectral theory
What is spectral theory?
OPs
OPRL basics Examples include Favard's Theorem Can you hear the shape of a drum ? OPUC basics
Szeg® recursion Computer tomography and Verblunsky coe cients Isospectral manifold for the harmonic oscillator BernsteinSzeg® Approximation
Carmona Simon Formula What is spectral theory
What is spectral theory?
OPs The direct problem goes from the object to spectra. OPRL basics The inverse problem goes backwards. Favard's Theorem The direct problem is typically easy while the inverse OPUC basics problem is typically hard. Szeg® recursion and Verblunsky coe cients For example, the domain of de nition of the harmonic
BernsteinSzeg® oscillator isospectral manifold is unknown. It is not even Approximation known if it is connected! Carmona Simon Formula OPs
What is spectral theory? OPs Orthogonal polynomials on the real line (OPRL) and on the OPRL basics unit circle (OPUC) are particularly useful because the Favard's Theorem inverse problems are easyindeed the inverse problem is the OPUC basics OP de nition as we'll see. Szeg® recursion and Verblunsky coe cients OPs also enter in many applicationboth speci c
BernsteinSzeg® polynomials and the general theory. Approximation
Carmona Simon Formula OPs
What is spectral theory? OPs Indeed, my own interest came from studying discrete OPRL basics Schrödinger operators on `2( ) Favard's Theorem Z OPUC basics Hu = un+1 + un−1 + V un Szeg® recursion n and Verblunsky coe cients and the realization that when restricted to Z+, one had a BernsteinSzeg® special case of OPRL. Approximation
Carmona Simon Formula OPRL basics
µ will be a probability measure on R. We'll always suppose What is spectral that has bounded support which is not a nite set of theory? µ [a, b]
OPs points. (We then say that µ is non-trivial.) This implies 2 OPRL basics that 1, x, x ,... are independent since R 2 Favard's Theorem |P (x)| dµ = 0 ⇒ µ is supported on the zeroes of P . OPUC basics Apply Gram Schmidt to 1, x, . . . and get monic polynomials Szeg® recursion and Verblunsky coe cients j j−1 Pj(x) = x + αj,1x + ... BernsteinSzeg® Approximation
Carmona Simon and orthonormal (ON) polynomials Formula
pj = Pj/kPjk OPRL basics
More generally we can do the same for any probability What is spectral theory? measure of bounded support on C. OPs One dierence from the case of , the linear combination of OPRL basics R {xj}∞ are dense in L2( , dµ) by Weierstrass. This may or Favard's Theorem j=0 R may not be true if supp . OPUC basics (dµ) 6⊂ R If on , the span of j ∞ is not dense in Szeg® recursion dµ = dθ/2π ∂D {z }j=0 and Verblunsky 2 2 coe cients L (but is only H ). Perhaps, surprisingly, we'll see later BernsteinSzeg® that there are measures dµ on ∂D for which they are dense Approximation (e.g., purely singular). Carmona Simon µ Formula More signi cantly, the argument we'll give for our recursion relation fails if supp(dµ) 6⊂ R. OPRL basics
Since is monic and k+1 span polynomials of degree Pk {Pj}j=0 What is spectral at most k + 1, we have theory? k X OPs xPk = Pk+1 + Bk,j Pj OPRL basics j=0 Favard's Theorem Clearly OPUC basics 2 Szeg® recursion Bk,j = hPj, xPki/kPjk and Verblunsky coe cients Now we use BernsteinSzeg® Approximation hPj, xPki = hxPj,Pki Carmona Simon (need on !!) Formula dµ R If j < k − 1, this is zero. 2 If j = k − 1, hPk−1, xPki = hxPk−1,Pki = kPkk . OPRL basics
Thus ( ); ∞ , ∞ : Jacobi recursion What is spectral P−1 ≡ 0 {aj}j=1 {bj}j=1 theory? OPs 2 xPN = PN+1 + bN+1PN + aN PN−1 OPRL basics
Favard's Theorem bN ∈ R, aN = kPN k/kPN−1k OPUC basics
Szeg® recursion These are called Jacobi parameters. This implies and Verblunsky coe cients kPN k = aN aN−1 . . . a1 (since kP0k = 1). BernsteinSzeg® Approximation This, in turn, implies pn = Pn/a1 . . . an obeys Carmona Simon Formula xpn = an+1pn+1 + bn+1pn + anpn−1 OPRL basics
We have thus solved the inverse problem, i.e., is the What is spectral µ theory? spectral data and ∞ are the descriptors of the {an, bn}n=1 OPs object. OPRL basics In the orthonormal basis ∞ , multiplication by has Favard's Theorem {pn}n=0 x the matrix OPUC basics Szeg® recursion and Verblunsky b1 a1 0 0 ... coe cients a1 b2 a2 0 ... BernsteinSzeg® Approximation J = 0 a b a ... 2 3 3 Carmona Simon ...... Formula . . . . . called a Jacobi matrix. Favard's Theorem
Since Z Z What is spectral 2 theory? bn = xpn−1(x) dµ, an = xpn−1(x)pn(x) dµ OPs
OPRL basics supp(µ) ⊂ [−R,R] ⇒ |bn| ≤ R, |an| ≤ R. Favard's Theorem Conversely, if , is a bounded OPUC basics supn |an| + |bn| = α < ∞ J matrix of norm at most . In that case, the spectral Szeg® recursion 3α and Verblunsky theorem implies there is a measure so that coe cients dµ BernsteinSzeg® Z Approximation h(1, 0,...)t,J `(1, 0,...)ti = x`dµ(x) Carmona Simon Formula If one uses Gram-Schmidt to orthonormalize ` t ∞ , one nds has Jacobi matrix exactly {J (1, 0,...) }`=0 µ given by J. Favard's Theorem
We have thus proven Favard's Theorem (his paper was in 1935; really due to Stieltjes in 1894 or to Stone in 1932). What is spectral theory? Favard's Theorem.There is a oneone correspondence
OPs between bounded Jacobi parameters
OPRL basics ∞ ∞ Favard's Theorem {an, bn}n=1 ∈ (0, ∞) × R OPUC basics
Szeg® recursion and non-trivial probability measures, µ, of bounded support and Verblunsky coe cients via: µ ⇒ {a , b } (OP recursion) BernsteinSzeg® n n Approximation {a , b } ⇒ µ (Spectral Theorem) Carmona Simon n n Formula There are also results for µ's with unbounded support so R n long as x dµ < ∞. In this case, {an, bn} ⇒ µ may not be unique because J may not be essentially self-adjoint on vectors of nite support. OPUC basics
What is spectral theory? OPs Let dµ be a non-trivial probability measure on ∂D. As in OPRL basics the OPRL case, we use Gram-Schmidt to de ne monic OPs, Favard's Theorem Φn(z) and ON OP's ϕn(z). OPUC basics Szeg® recursion In the OPRL case, if z is multiplication by the underlying and Verblunsky ∗ ∗ coe cients variable, z = z. This is replaced by z z = 1.
BernsteinSzeg® Approximation In the OPRL case, Pn+1 − xPn ⊥ {1, x1, . . . , xn−2}. Carmona Simon Formula OPUC basics
What is spectral n theory? In the OPUC case, Φn+1 − zΦn ⊥ {z, . . . , z }, since OPs j j−1 OPRL basics hzΦ, z i = hΦ, z i Favard's Theorem OPUC basics if j ≥ 1. Szeg® recursion and Verblunsky In the OPRL case, we used deg P = n and coe cients n−2 P ⊥ {1, x, . . . , x } ⇒ P = c1Pn + c2Pn−1. BernsteinSzeg® Approximation In the OPUC case, we want to characterize deg P = n, Carmona Simon 2 n Formula P ⊥ {z, z , . . . , z }. OPUC basics
De ne ∗ on degree n polynomials to themselves by
1 What is spectral Q∗(z) = zn Q theory? z¯ OPs
OPRL basics (bad but standard notation!) or
Favard's Theorem n n OPUC basics X j ∗ X j Q(z) = cjz ⇒ Q (z) = cn−j z Szeg® recursion and Verblunsky j=0 j=0 coe cients BernsteinSzeg® Then, ∗ is unitary and so for deg Q = n Approximation Carmona Simon n−1 Formula Q ⊥ {1, . . . , z } ⇔ Q = c Φn is equivalent to
n ∗ Q ⊥ {z, . . . , z } ⇔ Q = c Φn Szeg® recursion and Verblunsky coe cients
Thus, we see, there are parameters ∞ (called {αn}n=0 Verblunsky coe cients) so that What is spectral theory? ∗ OPs Φn+1(z) = zΦn − αnΦn(z) OPRL basics
Favard's Theorem This is the Szeg® Recursion (History: Szeg® and Geronimus
OPUC basics in 1939; Verblunsky in 193536) Szeg® recursion Applying ∗ for deg polynomials to this yields and Verblunsky n + 1 coe cients ∗ ∗ BernsteinSzeg® Φ (z) = Φ (z) − αnzΦn Approximation n+1 n
Carmona Simon Formula The strange looking −α¯n rather than say +αn is to have the αn be the Schur parameter of the Schur function of µ (Geronimus); also the Verblunsky parameterization then
agrees with αn. These are discussed in [OPUC1]. Szeg® recursion and Verblunsky coe cients
What is spectral theory? OPs monic constant term in ∗ is 1 ∗ . Φn ⇒ Φn ⇒ Φn(0) = 1 OPRL basics Favard's Theorem This plus ∗ implies Φn+1 = zΦn − α¯nΦn(z) OPUC basics
Szeg® recursion and Verblunsky −Φn+1(0) = αn coe cients BernsteinSzeg® i.e., determines . Approximation Φn αn−1
Carmona Simon Formula Szeg® recursion and Verblunsky coe cients
For OPRL, we saw kPn+1k/kPnk = an+1. We are looking for the analog for OPUC. What is spectral theory? Szeg® Recursion ∗ ⇒ Φn+1 +α ¯nΦn = zΦn OPs
OPRL basics ∗ 2 2 ∗ 2 2 Favard's Theorem Φn+1 ⊥ Φn ⇒ kΦn+1k + |αn| kΦnk = kzΦnk OPUC basics
Szeg® recursion and Verblunsky ∗ coe cients Multiplication by z unitary plus antiunitary ⇒ BernsteinSzeg® Approximation 2 2 2 2 2 kΦn+1k = ρn kΦnk ; ρn = 1 − |αn| Carmona Simon Formula which implies |αn| < 1 (i.e., αn ∈ D) and
kΦnk = ρn−1 ··· ρ0 Szeg® recursion and Verblunsky coe cients
What is spectral theory? OPs ϕn+1 ϕn −1 z −α¯n ∗ = An(z) ∗ x; An = ρn OPRL basics ϕn+1 ϕn −αn z 1 Favard's Theorem
OPUC basics Szeg® recursion det An 6= 0 if z 6= 0, so we can get ϕn (Φn) from ϕn+1 and Verblunsky coe cients (Φn+1) by
BernsteinSzeg® Approximation −2 ∗ zΦn = ρn Φn+1 +α ¯nΦn+1 Carmona Simon Formula ∗ −2 Φn = ρn Φn+1 + αnΦn+1 Szeg® recursion and Verblunsky coe cients
What is spectral theory?
OPs OPRL basics We see that Φn+1 determines αn, so by induction and Favard's Theorem inverse recursion, OPUC basics Szeg® recursion Theorem. If two measures have the same Φ , they have and Verblunsky n coe cients the same n−1 and n−1. {Φj}j=0 {αj}j=0 BernsteinSzeg® Approximation
Carmona Simon Formula Szeg® recursion and Verblunsky coe cients
What is spectral theory? A similar argument to the one that led to |αn| < 1 yields OPs Theorem. All zeros of lie in . OPRL basics Φn D Favard's Theorem OPUC basics Proof. Φn(z0) = 0 ⇒ Φn = (z − z0)p, deg p = n − 1 Szeg® recursion 2 2 2 2 and Verblunsky zp = Φn + z0p and p ⊥ Φn ⇒ kpk = kΦnk + |z0| kpk coe cients BernsteinSzeg® ⇒ |z0| < 1 Approximation Carmona Simon Corollary. All zeros of ∗ lie in . Formula Φn(z) C \ D Szeg® recursion and Verblunsky coe cients
What is spectral theory? Here is a second proof that only uses Szeg® recursion. By OPs induction, suppose that all zeros of Φn are in D. Then, for OPRL basics |β| < 1 Favard's Theorem ∗ on zΦn + βΦn 6= 0 ∂D OPUC basics since ∗ on .( 1 ) Szeg® recursion |zΦn(z)| = |Φn(z)| ∂ = z and Verblunsky D z¯ coe cients If (β) ∗ , then at , all zeros of (β) are BernsteinSzeg® Φn+1 = zΦn + βΦn β = 0 Φn+1 Approximation in D. Carmona Simon (β) Formula As varies in , all zeros of are trapped in . QED. β D Φn+1 D BernsteinSzeg® Approximation
We are heading towards a proof that any ∞ are {αn}n=0 ⊂ D the Verblunsky coe cients of a measure on ∂D (analog of What is spectral Favard's Theorem). It will depend on theory? OPs Theorem (BernsteinSzeg® measures). Let OPRL basics (0) n−1 n. Let be the normalized degree Favard's Theorem {αj }j=0 ∈ D ϕn(z) n polynomial obtained by Szeg® recursion. Let OPUC basics
Szeg® recursion and Verblunsky dθ coe cients dµn(θ) = iθ 2 2π|ϕn(e )| BernsteinSzeg® Approximation
Carmona Simon Formula Then dµn has Verblunsky coe cients
( (0) αj j = 0, . . . , n − 1 αj(dµn) = 0 j ≥ n BernsteinSzeg® Approximation
The rst step of the proof is to show that
What is spectral theory? Z k ` OPs k, `, n with k < n + ` ⇒ z¯ z ϕn(z)dµn(θ) = 0 z=eiθ OPRL basics
Favard's Theorem For 1 −n ∗ . OPUC basics z ∈ ∂D ⇒ ϕn(z) = ϕn z¯ = z ϕn(z) Szeg® recursion Thus the integral above is and Verblunsky coe cients I k ` I BernsteinSzeg® z¯ z ϕn(z) dz 1 `+n−k−1 dz Approximation = z z−n ϕ (z) ϕ∗ (z) 2πiz 2π ϕ∗ (z) Carmona Simon n n n Formula is zero since ∗ −1 is analytic on a neighborhood of ϕn(z) D and ` + n − k − 1 ≥ 0. BernsteinSzeg® Approximation
` Thus, z ϕn is a multiple of the OP's for dµn. What is spectral theory? R ` 2 Since |z ϕn| dµ = 1, we see that OPs OPRL basics k ϕn+k(z; dµ) = z ϕn(z); k > 0. Favard's Theorem OPUC basics As we saw, determines n−1 and by inverse Φn {αj}j=0 Φj Szeg® recursion and Verblunsky Szeg® recursion and −Φ (0) = α . This shows that coe cients j+1 j BernsteinSzeg® ( Approximation ϕ (x) j = 0, . . . , n ϕ (z; dµ) = j Carmona Simon j j−n Formula z ϕn(z) j = n, n + 1,... implying the claimed result. BernsteinSzeg® Approximation
Given ∞ ∞, we can form as above. Via {αj}j=0 ⊂ D dµn What is spectral R Φ (eiθ)dµ(eiθ) = 0, {Φ }n determines {R zjdµ}n theory? j j j=0 j=0 inductively (actually they determine more moments). Thus OPs OPRL basics Z Z j j Favard's Theorem z dµn = z dµm j ≤ min(n, m) OPUC basics
Szeg® recursion and Verblunsky and R zj dµ = R zj dµ . coe cients n n BernsteinSzeg® Thus, dµn has a weak limit dµ∞. Clearly, αj(dµ∞) = αj. Approximation We have thus proven Carmona Simon Formula Verblunsky's Theorem. ∞ sets up a 11 µ → {αj(µ)}j=0 correspondence between non-trivial probability measures on ∞ ∂D and D . Carmona Simon Formula
Simon [CRM Proc. and Lecture Notes 42 (2007), 453463] What is spectral has proven an analog of the BernsteinSzeg® approximation theory? for OPRL (the analog for Schrödinger operators is due to OPs
OPRL basics Carmona; hence the name): R n Favard's Theorem Let dρ be a probability measure on R with |x| dρ < ∞ OPUC basics for all . Let ∞ be its orthonormal polynomials and n {pn}n=0 Szeg® recursion ∞ its Jacobi parameters. Let and Verblunsky {an, bn}n=1 coe cients BernsteinSzeg® dν (x) = dx/π(a2 p2 (x) + p2 (x)) Approximation n n n n−1
Carmona Simon R ` R ` Formula Then, for ` = 0,..., 2n − 2, x dνn = x dρ. If the moment problem for dρ is determinate, then dνn → dρ weakly. Carmona Simon Formula
One important consequence of this result is
What is spectral Theorem. If is an interval and for all and theory? I ⊂ R x ∈ I some , we have that OPs c > 0 2 2 2 −1 OPRL basics c ≤ anpn(x) + pn−1(x) ≤ c
Favard's Theorem then dρ I has a.c. part and no singular spectrum. OPUC basics Similarly, for and a probability measure Szeg® recursion I ⊂ ∂D µ and Verblunsky coe cients −1 c ≤ |ϕn(z)| ≤ c all z ∈ I BernsteinSzeg® Approximation implies has a.c. part and no singular spectrum. Carmona Simon dµ I Formula Remark. A much stronger result is known (see e.g., Simon [Proc AMS 124 (1996), 3361]); I can be any set and c can be x-dependent.