Inversion Problem with Separable Potentials

Abstract

The problem of finding a potential, expressible as the sum of a finite number of separable terms, to fit a given phase shift at all energies is solved quite generally for nonrelativistic scattering within a single channel. The most general solution is found, and the necessary restrictions on the phase shift and the role of bound states are studied. The minimum number of separable terms needed to fit a given phase shift in a given channel is found to be determined, normally, by $$\text{1+}\max {\displaystyle {lim}_{x_2{\mathrm{>x}}_1}}{\mathrm{\{[\delta (x}}_2{\mathrm{)/\pi ]-[\delta (x}}_1\mathrm{)/\pi ]\}}$$ , where x₁ and x₂ range through all positive energies, and [y] is the largest integer less than or equal to y. Our method differs from that of Kh. Chadan [Nuovo Cimento 10, 892 (1958); 47, 510 (1967)] who has solved what is, in essence, the same problem, in that it involves spherical Bessel transforms throughout, rather than the use at each stage of a representation determined by the solution of a previous stage.