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

In this paper, we study the small‐λ$\lambda$ spectral asymptotics of an integral operator K$\mathcal {K}$ defined on two multi‐intervals J$J$ and E$E$, when the multi‐intervals touch each other (but their interiors are disjoint). The operator K$\mathcal {K}$ 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 dist(J,E)>0$\text{dist}(J,E)>0$, and K$\mathcal {K}$ is of the Hilbert–Schmidt class, was studied in an earlier paper by the authors. The main assumption in this paper is that U=J∪E$U=J\cup E$ is a single interval (although part of our analysis is valid in a more general situation). We show that the eigenvalues of K$\mathcal {K}$, if they exist, do not accumulate at λ=0$\lambda =0$. Combined with the results in an earlier paper by the authors, this implies that Hp$H_p$, the subspace of discontinuity (the span of all eigenfunctions) of K$\mathcal {K}$, is finite dimensional and consists of functions that are smooth in the interiors of J$J$ and E$E$. We also obtain an approximation to the kernel of the unitary transformation that diagonalizes K$\mathcal {K}$, and obtain a precise estimate of the exponential instability of inverting K$\mathcal {K}$. Our work is based on the method of Riemann–Hilbert problem and the nonlinear steepest descent method of Deift and Zhou.