Spectral edge behavior for eventually monotone Jacobi and Verblunsky\n coefficients

2017/05/26 by Milivoje Lukić, Lukic, Milivoje · 1 citation
Mathematics · Physics and Astronomy · #39A70 (Secondary) #42C05 #47B36 (Primary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1705.09461

openalex publication_date 2017/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Jacobi matrices with eventually increasing sequences of diagonal\nand off-diagonal Jacobi parameters. We describe the asymptotic behavior of the\nsubordinate solution at the top of the essential spectrum, and the asymptotic\nbehavior of the spectral density at the top of the essential spectrum.\n In particular, allowing on both diagonal and off-diagonal Jacobi parameters\nperturbations of the free case of the form - \∑_j=1^J c_j n^-\τ_j +\no(n^-\τ₁-1) with 0 < \τ₁ < \τ₂ < \… < \τ_J and c₁>0, we\nfind the asymptotic behavior of the \log of spectral density to order\nO(\log(2-x)) as x approaches 2.\n Apart from its intrinsic interest, the above results also allow us to\ndescribe the asymptotics of the spectral density for orthogonal polynomials on\nthe unit circle with real-valued Verblunsky coefficients of the same form.\n

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