Suppose that for a potential u(x) for the
Schrödinger operator L = -\frac{d^2}{dx^2} + u(x), one has the
scattering data (r(k), \{\chi_1, \cdots, \chi_N\}), where r(k) are the reflection coefficients from continuous scattering, given as a function r: \mathbb{R} \rightarrow \mathbb{C}, and the real parameters \chi_1, \cdots, \chi_N > 0 are from the discrete bound spectrum. Then defining F(x) = \sum_{n=1}^N\beta_ne^{-\chi_n x} + \frac{1}{2\pi} \int_\mathbb{R}r(k)e^{ikx}dk, where the \beta_n are non-zero constants, solving the GLM equation K(x,y) + F(x+y) + \int_x^\infty K(x,z) F(z+y) dz = 0 for K allows the potential to be recovered using the formula u(x) = -2 \frac{d}{dx}K(x,x). == See also ==