Sum rules for Jacobi matrices and their applications to spectral theory

By Rowan Killip, Barry Simon. 2001

285 citations recorded by the source corpus.

We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J - J(0) is Hilbert-Schmidt, and a proof of Nevai's conjecture that the Szego condition holds if J - J(0) is trace class.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d8f986a68eac3b87f3fc851e5b74cc2f7bf23677

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Orthogonal polynomials on the unit circle: New results (2004, unknown): “Interestingly enough, while the proof in [69] has an easy half that depends on semicontinuity of the entropy and a hard half (that even after simplifications in [129, 123] is not so short), the analog of the hard half for (2.1) follows in a few lines from Jensen's inequality and goes back to Szeg˝…”
FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS, I. A TALE OF TWO PICTURES (2004, methodology): “I will follow ideas of Killip-Simon [20] and Damanik-Simon [7, 8], as discussed in [35, Chapter 13].”
Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems (2005, background): “Since ZQ is an entropy [9, 17], it is upper semi-continuous with respect to weak convergence of measures, and so ZQ(μ) ≥ lim supn ZQ(μn) with μn the BernsteinSzegő approximations of μ having Verblunsky coefficients {α0, .”
Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics (2005, methodology): “The first two Taylor terms at 0 yield (β)–(γ), and as in [26], a suitable combination of the first and third Taylor coefficients is positive and yields (α).”
Orthogonal polynomials (2005, unknown): “After numerous works in the subject by Szegő, Shohat, Geronimus, Krein, Kolmogorov and others, a complete characterization for J − J0 being a HilbertSchmidt operator was given in [43] by R.”
CMV matrices: Five years after (2006, methodology): “For Jacobi matrices, under the name “perturbation determinant,” they were used in [52].”
Fine Structure of the Zeros of Orthogonal Polynomials, IV. A Priori Bounds and Clock Behavior (2006, unknown): “iate. Proof of Theorem 8.9. By scaling, we suppose I= [−2,2]. By Corollary A.3, Kn(x,x;dµ) ≤ Kn(x,x;dν) (8.37) where dν= χ[−2,2]wdx (8.38) Let d˜ν be the normalized dν. By the theorem of Killip-Simon [25], the Jacobi parameters obey (8.35), so by (8.37), sup x∈[−2+δ,2−δ] Kn(x,x;dµ) ≤ exp Cδ √ n+1 In”
Sum rules and spectral measures of Schrödinger operators with L^2 potentials (2006, methodology): “This is actually the natural independent variable for what follows (in [20] we used z defined by E = z + z−1).”
The Analytic Theory of Matrix Orthogonal Polynomials (2007, methodology): “Following Killip–Simon [128] and Simon [167, 168, 176], our Jacobi matrices are indexed with n = 1, 2, .”