Integral equation methods for the inverse problem with discontinuous wave speed

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Abstract
The recovery of the coefficient H(x) in the one‐dimensional generalized Schrödinger equation d2ψ/dx2+k2H(x)2ψ=Q(x)ψ, where H(x) is a positive, piecewise continuous function with positive limits H± as x→±∞, is studied. The large‐k asymptotics of the wave functions and the scattering coefficients are analyzed. A factorization formula is given expressing the total scattering matrix as a product of simpler scattering matrices. Using this factorization an algorithm is presented to obtain the discontinuities in H(x) and H′(x)/H(x) in terms of the large‐k asymptotics of the reflection coefficient. When there are no bound states, it is shown that H(x) is recovered from an appropriate set of scattering data by using the solution of a singular integral equation, and the unique solvability of this integral equation is established. An equivalent Marchenko integral equation is derived and is shown to be uniquely solvable; the unique recovery of H(x) from the solution of this Marchenko equation is presented. Some explicit examples are given, illustrating the recovery of H(x) from the solution of the singular integral equation and from that of the Marchenko equation.