ar X iv : m at h - ph / 0 10 20 28 v 1 2 2 Fe b 20 01 Reconstruction of the potential from I - function . ∗ †

If f (x, k) is the Jost solution and f (x) = 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), k j , s j , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = − d 2 dx 2 + q(x) on (0, ∞) with the Dirichlet condition at x = 0 and q(x) ∈ L 1,1 := {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).