Application of scattering theory to a problem in hydrodynamics

The linearized equations of motion of a layer of heavy perfect incompressible fluid above an irregular stratum are written as an equation of the type if-prime = Af in Hilbert space with a certain self-adjoint operator A. Scattering-theory methods are used to compute the continuous spectrum and describe the eigenfunctions of the operator A on the assumption that only a compact part of the irregular stratum differs from a horizontal plane. It is shown that the operator A is equivalent to an operator of the form k (+) (-k), where k is a nonnegative self-adjoint operator, and that solutions of the form X (+ or -) exist.