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-$\ell$ 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 $\ell$-problem, are fulfilled. The uniqueness is demonstrated using the JWKB approximation.