2020/04/23 by Zhaoyu Wang, Wang, Zhaoyu, Engui Fan +1 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #Differential Equations and Numerical Methods #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2004.11971
openalex publication_date 2020/04/23 · openalex created_date 2020/05/01 · openalex updated_date 2026/07/28
In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight wpJ2(x,t)=e-(t)/(x(1-x))xα(1-x)β, where t ≥ 0, α>0, β>0 and x ∈ [0,1]. Our main results obtained here include two aspects: I. Strong asymptotics: We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval (0,1) and outside of interval ℂ\backslash (0,1), respectively; Due to the effect of (t)/(x(1-x)) for varying t, different asymptotic behaviors at the hard edge 0 and 1 were found with different scaling schemes. Specifically, the uniform asymptotic behavior can be expressed as a Airy function in the neighborhood of point 1 as ζ= 2n2t → ∞, n→ ∞, while it is given by a Bessel function as ζ→ 0, n → ∞. II. Universality: We respectively calculate the limit of the eigenvalue correlation kernel in the bulk of the spectrum and at the both side of hard edge, which will involve a ψ-functions associated with a particular Painlev\acutee \uppercase\expandafter\romannumeral3 equation near x=± 1. Further, we also prove the ψ-funcation can be approximated by a Bessel kernel as ζ→ 0 compared with a Airy kernel as ζ→ ∞. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problems.