Construction of potentials using mixed scattering data

The long-standing problem of constructing a potential from mixed scattering data is discussed. We first consider the fixed-$\mathrm{\ell }$ inverse scattering problem. We show that the zeros of the regular solution of the Schr"odinger equation, $r_n(E)$ which are monotonic functions of the energy, determine a unique potential when the domain of energy is such that the $r_n(E)$'s range from zero to infinity. The latter method is applied to the domain ${E \geq E_0, \ell=\ell_0 } \cup {E=E_0, \ell \geq \ell_0 }$ for which the zeros of the regular solution are monotonic in both parts of the domain and still range from zero to infinity. Our analysis suggests that a unique potential can be obtained from the mixed scattering data ${\delta(\ell_0,k), k \geq k_0 } \cup {\delta(\ell,k_0), \ell \geq \ell_0 }$ provided that certain integrability conditions required for the fixed $\mathrm{\ell }$-problem, are fulfilled. The uniqueness is demonstrated using the JWKB approximation.