  We use inverse scattering methods, generalized for a specific class of
complex potentials, to construct a one parameter family of complex potentials
V(s, r) which have the property that the zero energy s-wave Jost function, as a
function of s alone, is identical to Riemann's $\xi$ function whose zeros are
the non-trivial zeros of the zeta function. These potentials have an asymptotic
expansion in inverse powers of s(s-1) with real coefficients V_n(r) which are
explicitly calculated. We show that the validity of the Riemann hypothesis
depends essentially on simple integrability properties of the first order
coefficient, V_1(r). In the case studied in this paper, this coefficient does
not satisfy these conditions, but proof of that fact does indicate several
possibilities for proceeding further.
