Inverse Scattering/ Inversion of Gamows Formula

Classically it is impossible for a particle to penetrate a potential barrier exceeding its energy. In quantum mechanics, however, there is always a finite probability for such an occurrence. The transmission probability is given by Gamow's formula: \begin{equation} T(E)=e^{-{2\over \hbar}\int_{x_1(E)}^{x_2(E)}{\sqrt{2m(V(x)-E)}\,\mathrm{d}x}}, \end{equation} where $V(x)$ is the potential, $E$ the particles energy, and $x_1$ and $x_2$ are the classical reflection points. $T(E)$ can typically be easily obtained through experiment. However, the task of determining $V(x)$ is often more difficult. Hence, the inversion of Gamow's formula would provide a tool for probing physical structure. Procedures for doing so and results obtained shall be discussed.