Reconstruction of the potential from I-function

Abstract

If $f(x,k)$ is the Jost solution and $f(x)=f(0,k)$, then the $I$-function is $I(k):=\frac{f^{\prime }(0,k)}{f(k)}$. It is proved that $I(k)$ is in one-to-one correspondence with the scattering triple $\mathcal{S}:=\{S(k),k_j,s_j,1\le j\le J\}$ and with the spectral function $\rho (\lambda )$ of the Sturm-Liouville operator $l=-\frac{d^2}{d{x}^2}+q(x)$ on $(0,\infty )$ with the Dirichlet condition at $x=0$ and $q(x)\in L_{1,1}:=\{q:q=\bar{q},{\int }_0^{\infty }(1+x)|q(x)dx<\infty \}$. Analytical methods are given for finding $\mathcal{S}$ from $I(k)$ and $I(k)$ from $\mathcal{S}$, and $\rho (\lambda )$ from $I(k)$ and $I(k)$ from $\rho (\lambda )$. Since the methods for finding $q(x)$ from $\mathcal{S}$ or from $\rho (\lambda )$ are known, this yields the methods for finding $q(x)$ from $I(k)$.