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 coecients 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 coecients 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, coecients 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 coecients 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 coecients 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 coecients For example, the domain of denition 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 denition as we'll see. Szeg® recursion and Verblunsky coecients OPs also enter in many applicationboth specic 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 coecients 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 jP (x)j 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 coecients 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 fxjg1 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 1 is not dense in Szeg® recursion dµ = dθ=2π @D fz gj=0 and Verblunsky 2 2 coecients 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 signicantly, 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 fPjgj=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 coecients 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 ( ); 1 , 1 : Jacobi recursion What is spectral P−1 ≡ 0 fajgj=1 fbjgj=1 theory? OPs 2 xPN = PN+1 + bN+1PN + aN PN−1 OPRL basics Favard's Theorem bN 2 R; aN = kPN k=kPN−1k OPUC basics Szeg® recursion These are called Jacobi parameters. This implies and Verblunsky coecients 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 1 are the descriptors of the fan; bngn=1 OPs object. OPRL basics In the orthonormal basis 1 , multiplication by has Favard's Theorem fpngn=0 x the matrix OPUC basics Szeg® recursion 0 1 and Verblunsky b1 a1 0 0 ::: coecients Ba1 b2 a2 0 :::C BernsteinSzeg® B C Approximation J = B 0 a b a :::C @ 2 3 3 A 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] ) jbnj ≤ R; janj ≤ R. Favard's Theorem Conversely, if , is a bounded OPUC basics supn janj + jbnj = α < 1 J matrix of norm at most . In that case, the spectral Szeg® recursion 3α and Verblunsky theorem implies there is a measure so that coecients 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 1 , one nds has Jacobi matrix exactly fJ (1; 0;:::) g`=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 1 1 Favard's Theorem fan; bngn=1 2 (0; 1) × R OPUC basics Szeg® recursion and non-trivial probability measures, µ, of bounded support and Verblunsky coecients via: µ ) fa ; b g (OP recursion) BernsteinSzeg® n n Approximation fa ; b g ) 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 < 1. In this case, fan; bng ) 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 dene monic OPs, Favard's Theorem Φn(z) and ON OP's 'n(z). OPUC basics Szeg® recursion In the OPRL case, if z is multiplication by the underlying and Verblunsky ∗ ∗ coecients variable, z = z. This is replaced by z z = 1. BernsteinSzeg® Approximation In the OPRL case, Pn+1 − xPn ? f1; x1; : : : ; xn−2g. Carmona Simon Formula OPUC basics What is spectral n theory? In the OPUC case, Φn+1 − zΦn ? fz; : : : ; z g, 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 coecients n−2 P ? f1; x; : : : ; x g ) P = c1Pn + c2Pn−1. BernsteinSzeg® Approximation In the OPUC case, we want to characterize deg P = n, Carmona Simon 2 n Formula P ? fz; z ; : : : ; z g. OPUC basics Dene ∗ 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 coecients BernsteinSzeg® Then, ∗ is unitary and so for deg Q = n Approximation Carmona Simon n−1 Formula Q ? f1; : : : ; z g , Q = c Φn is equivalent to n ∗ Q ? fz; : : : ; z g , Q = c Φn Szeg® recursion and Verblunsky coecients Thus, we see, there are parameters 1 (called fαngn=0 Verblunsky coecients) 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 coecients ∗ ∗ 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.