2015/09/30 by Thomas Bothner · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.PR #math.SP #nlin.SI #msc:45C05 #msc:45M05 #msc:82B26 #msc:33C10 #msc:33C45
paper · pdf · doi:10.1088/1751-8113/49/7/075204
50 pages, 10 figures. To appear in J. Phys. A: Mathematical and Theoretical. Version 2 corrects typos and updates literature
arxiv created 2016/01/13 · arxiv updated 2016/02/17
We present a method to derive asymptotics of eigenvalues for trace-class integral operators K:L2(J;dλ)\circlearrowleft, acting on a single interval J⊂ℝ, which belong to the ring of integrable operators \citeIIKS. Our emphasis lies on the behavior of the spectrum \λi(J)\i=0∞ of K as |J|→∞ and i is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant det(I-γK)|L2(J) as |J|→∞ and γ\uparrow 1 in a Stokes type scaling regime. Concrete asymptotic formulæ are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory.