On the inverse and determinant of certain truncated Wiener--Hopf operators (Q877859)

The solution of a problem arising in integrable systems [cf.\ \textit{H.\,Widom}, ``Asymptotics of a class of operator determinants with application to the cylindrical Toda equations'' (Preprint) (2006; arXiv:nlin.SI/0605021)] requires the asymptotics as \(\alpha\to\infty\), with very small error, of the inverses and determinants of truncated Wiener--Hopf operators \(W_\alpha(\sigma)\) acting on \(L^2(0,\alpha)\), both in the regular case (where the Wiener--Hopf operator \(W(\sigma)\) on \(L^2({\mathbb R}_+)\) is invertible) and in singular cases. The author considers singular symbols of the form \(\sigma(\xi)=(\xi^2-p^2)\tau(\xi)\) (Fisher--Hartwig symbols), where \(p\) is real and \(\tau\) is nonzero, has index zero, and its Wiener--Hopf factors are of order \(1/\xi\) as \(\xi\to\infty\). Asymptotic formulas for \(W_\alpha(\sigma)^{-1}\) and \(\det W_\alpha(\sigma)\) are obtained, both with error \(O(e^{-\delta\alpha})\) for some \(\alpha>0\). These formulas involve Hankel operators with distribution symbols.