Reconstruction of the potential from I-function

If f(x, k) is the Jost solution and f(k) = f(0, k), then the I-function is I(k) := f ′(0, k)/f(k). It is proved that I(k) is in one-to-one correspondence with the scattering triple S := {S(k), kj , sj , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = −d/dx + q(x) on (0,∞) with the Dirichlet condition at x = 0 and q(x) ∈ L1,1 := {q | q = q, ∫∞ 0 (1 + x)|q(x)dx <∞}. Analytical methods are given for finding S from I(k) and I(k) from S, and ρ(λ) from I(k) and I(k) from ρ(λ). Since the methods for finding q(x) from S or from ρ(λ) are known, this yields the methods for finding q(x) from I(k).