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Spectral Theory of Orthogonal Polynomials Floquet Solutions Spectral Theory of Orthogonal Polynomials Floquet Solutions Periodic Jacobi Periodic and Ergodic Spectral Problems Matrices Issac Newton Institute, January, 2015 The Discriminant Gaps Spectrum Barry Simon Potential Theory IBM Professor of Mathematics and Theoretical Physics Quadratic Irrationalities California Institute of Technology Minimal Herglotz Pasadena, CA, U.S.A. Functions Isospectral Tori Lecture 5: Isospectral Tori Spectral Theory of Orthogonal Polynomials Lecture 1: Introduction and Overview Lecture 2: Szegő Theorem for OPUC Floquet Solutions Periodic Jacobi Lecture 3: Three Kinds of Polynomial Asymptotics Matrices Lecture 4: Potential Theory The Discriminant Gaps Lecture 5: Isospectral Tori Spectrum Lecture 6: Fuchsian Groups Potential Theory Lecture 7: Chebyshev Polynomials, I Quadratic Irrationalities Lecture 8: Chebyshev Polynomials, II Minimal Herglotz Functions Isospectral Tori References [OPUC] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series Floquet Solutions 54.1, American Mathematical Society, Providence, RI, Periodic Jacobi Matrices 2005. The Discriminant [OPUC2] B. Simon, Orthogonal Polynomials on the Unit Gaps Circle, Part 2: Spectral Theory, AMS Colloquium Series, Spectrum 54.2, American Mathematical Society, Providence, RI, Potential Theory Quadratic 2005. Irrationalities Minimal Herglotz [SzThm] B. Simon, Szegő’s Theorem and Its Descendants: Functions Spectral Theory for L2 Perturbations of Orthogonal Isospectral Tori Polynomials, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011. Since you’ve seen some of these ideas already, I’ll be brief on some of the details. 1 So fan; bngn=−∞ are two-sided sequences with some p > 0 in Z so that an+p = an bn+p = bn 1 For z 2 C fixed, we are interested in solutions fungn=0 of anun+1 + bnun + an−1un−1 = zun Floquet Solutions We turn next to two sided periodic Jacobi matrices, their isospectral tori and use that as a jump off to the start of Floquet Solutions finite gap isospectral tori which will involve some almost Periodic Jacobi Matrices periodic examples. The Discriminant Gaps Spectrum Potential Theory Quadratic Irrationalities Minimal Herglotz Functions Isospectral Tori 1 For z 2 C fixed, we are interested in solutions fungn=0 of anun+1 + bnun + an−1un−1 = zun Floquet Solutions We turn next to two sided periodic Jacobi matrices, their isospectral tori and use that as a jump off to the start of Floquet Solutions finite gap isospectral tori which will involve some almost Periodic Jacobi Matrices periodic examples. Since you’ve seen some of these ideas The Discriminant already, I’ll be brief on some of the details. Gaps 1 So fan; bng are two-sided sequences with some p > 0 Spectrum n=−∞ in so that Potential Theory Z Quadratic an+p = an bn+p = bn Irrationalities Minimal Herglotz Functions Isospectral Tori Floquet Solutions We turn next to two sided periodic Jacobi matrices, their isospectral tori and use that as a jump off to the start of Floquet Solutions finite gap isospectral tori which will involve some almost Periodic Jacobi Matrices periodic examples. Since you’ve seen some of these ideas The Discriminant already, I’ll be brief on some of the details. Gaps 1 So fan; bng are two-sided sequences with some p > 0 Spectrum n=−∞ in so that Potential Theory Z Quadratic an+p = an bn+p = bn Irrationalities 1 Minimal Herglotz For z 2 C fixed, we are interested in solutions fungn=0 of Functions Isospectral Tori anun+1 + bnun + an−1un−1 = zun Such solutions are called Floquet solutions as they are analogs of solutions of ODE, especially Hill’s equation −u00 + V u = zu, V (x + p) = V (x). The analysis of such solutions is a delightful amalgam of three tools, the first of which is just the fact that the set of all solutions of the difference equation is two-dimensional. Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1: Floquet Solutions iθ that also obey for some λ 2 C (λ = e ; θ 2 C) Floquet Solutions un+p = λun Periodic Jacobi Matrices The Discriminant Gaps Spectrum Potential Theory Quadratic Irrationalities Minimal Herglotz Functions Isospectral Tori The analysis of such solutions is a delightful amalgam of three tools, the first of which is just the fact that the set of all solutions of the difference equation is two-dimensional. Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1: Floquet Solutions iθ that also obey for some λ 2 C (λ = e ; θ 2 C) Floquet Solutions un+p = λun Periodic Jacobi Matrices Such solutions are called Floquet solutions as they are The Discriminant analogs of solutions of ODE, especially Hill’s equation Gaps 00 Spectrum −u + V u = zu, V (x + p) = V (x). Potential Theory Quadratic Irrationalities Minimal Herglotz Functions Isospectral Tori the first of which is just the fact that the set of all solutions of the difference equation is two-dimensional. Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1: Floquet Solutions iθ that also obey for some λ 2 C (λ = e ; θ 2 C) Floquet Solutions un+p = λun Periodic Jacobi Matrices Such solutions are called Floquet solutions as they are The Discriminant analogs of solutions of ODE, especially Hill’s equation Gaps 00 Spectrum −u + V u = zu, V (x + p) = V (x). Potential Theory The analysis of such solutions is a delightful amalgam of Quadratic Irrationalities three tools, Minimal Herglotz Functions Isospectral Tori Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1: Floquet Solutions iθ that also obey for some λ 2 C (λ = e ; θ 2 C) Floquet Solutions un+p = λun Periodic Jacobi Matrices Such solutions are called Floquet solutions as they are The Discriminant analogs of solutions of ODE, especially Hill’s equation Gaps 00 Spectrum −u + V u = zu, V (x + p) = V (x). Potential Theory The analysis of such solutions is a delightful amalgam of Quadratic Irrationalities three tools, the first of which is just the fact that the set of Minimal Herglotz all solutions of the difference equation is two-dimensional. Functions Isospectral Tori If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1: Floquet Solutions iθ that also obey for some λ 2 C (λ = e ; θ 2 C) Floquet Solutions un+p = λun Periodic Jacobi Matrices Such solutions are called Floquet solutions as they are The Discriminant analogs of solutions of ODE, especially Hill’s equation Gaps 00 Spectrum −u + V u = zu, V (x + p) = V (x). Potential Theory The analysis of such solutions is a delightful amalgam of Quadratic Irrationalities three tools, the first of which is just the fact that the set of Minimal Herglotz all solutions of the difference equation is two-dimensional. Functions Isospectral Tori Thus, there are, for z fixed, at most two different λ’s for which there is a solution. λ1λ2 = 1: Floquet Solutions iθ that also obey for some λ 2 C (λ = e ; θ 2 C) Floquet Solutions un+p = λun Periodic Jacobi Matrices Such solutions are called Floquet solutions as they are The Discriminant analogs of solutions of ODE, especially Hill’s equation Gaps 00 Spectrum −u + V u = zu, V (x + p) = V (x). Potential Theory The analysis of such solutions is a delightful amalgam of Quadratic Irrationalities three tools, the first of which is just the fact that the set of Minimal Herglotz all solutions of the difference equation is two-dimensional. Functions Isospectral Tori Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies Floquet Solutions iθ that also obey for some λ 2 C (λ = e ; θ 2 C) Floquet Solutions un+p = λun Periodic Jacobi Matrices Such solutions are called Floquet solutions as they are The Discriminant analogs of solutions of ODE, especially Hill’s equation Gaps 00 Spectrum −u + V u = zu, V (x + p) = V (x). Potential Theory The analysis of such solutions is a delightful amalgam of Quadratic Irrationalities three tools, the first of which is just the fact that the set of Minimal Herglotz all solutions of the difference equation is two-dimensional. Functions Isospectral Tori Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1: It is the finite Jacobi matrix with 1p and p1 matrix elements added: Jjj = bj;Jj j+1 = aj;Jj j−1 = aj−1 −1 J1p = apλ ;Jp1 = apλ 1 −1 If fungn=−∞ is a Floquet solution, u0 = λ up, p per,λ up+1 = λu1 so ue = fungn=1 has J ue = zue. Conversely, if ue solves this, the unique u with un+p = λun 1 and ue = fungn=1 is a Floquet solution. Periodic B.C. Jacobi Matrices The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p.
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