Asymptotics of Determinants of Bessel Operators

 For aL∞(ℝ+)∩L1(ℝ+) the truncated Bessel operator Bτ(a) is the integral operator acting on L2[0,τ] with the kernel \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} where Jν stands for the Bessel function with ν>−1. In this paper we determine the asymptotics of the determinant det(I+Bτ(a)) as τ→∞ for sufficiently smooth functions a for which a(x)≠1 for all x[0,∞). The asymptotic formula is of the form det(I+Bτ(a))∼GτE with certain constants G and E, and thus similar to the well-known Szegö-Akhiezer-Kac formula for truncated Wiener-Hopf determinants.