A Jost--Pais theorem for the intermediate long wave equation

For a one-dimensional Schrödinger operator, the transmission coefficient can be expressed as a Fredholm determinant. This fundamental discovery of Jost and Pais has recently proven very influential in the study of conserved quantities for the Korteweg–de Vries equation, whose Lax operator takes this form. In this paper, we obtain an analogue of this result for the intermediate long wave equation, where, startlingly, the transmission coefficient is given by a \emph{ratio} of Fredholm determinants. We then recover the original Jost–Pais result in the shallow-water limit and its analogue for the Benjamin–Ono equation as the infinite-depth limit.