On singular limits of finite Hilbert transform operators on multi‐intervals

Abstract

AbstractIn this paper, we study the small‐ spectral asymptotics of an integral operator defined on two multi‐intervals and , when the multi‐intervals touch each other (but their interiors are disjoint). The operator is closely related to the multi‐interval finite Hilbert transform (FHT). This case can be viewed as a singular limit of self‐adjoint Hilbert–Schmidt integral operators with so‐called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when , and is of the Hilbert–Schmidt class, was studied in an earlier paper by the authors. The main assumption in this paper is that is a single interval (although part of our analysis is valid in a more general situation). We show that the eigenvalues of , if they exist, do not accumulate at . Combined with the results in an earlier paper by the authors, this implies that , the subspace of discontinuity (the span of all eigenfunctions) of , is finite dimensional and consists of functions that are smooth in the interiors of and . We also obtain an approximation to the kernel of the unitary transformation that diagonalizes , and obtain a precise estimate of the exponential instability of inverting . Our work is based on the method of Riemann–Hilbert problem and the nonlinear steepest descent method of Deift and Zhou.