Extrapolation of scattering data to the negative-energy region

Explicit analytic expressions are derived for the effective-range function for the case when the interaction is represented by a sum of the short-range square-well and long-range Coulomb potentials. These expressions are then transformed into forms convenient for extrapolating to the negative-energy region and obtaining the information about bound-state properties. Alternative ways of extrapolation are discussed. Analytic properties of separate terms entering these expressions for the effective-range function and the partial-wave scattering amplitude are investigated.

Figure 1

Functional dependence of $ik{C}_0^2\left(\eta \right)$ on energy $E$.

Figure 2

Functional dependence of $2k\eta h\left(\eta \right)$ on energy $E$.

Figure 3

Functional dependence of $k{C}_0^2\left(\eta \right)cot{\delta }_0$ on energy $E$.

Figure 4

Functional dependence of $k{C}_0^2\left(\eta \right)(cot{\delta }_0-i)$ on energy $E$.

Figure 5

Dependence of ERF $K_0\left(E\right)$ on energy $E$. The imaginary part is 0.

Figure 6

Functional dependence of $\mathrm{Re}[k{C}_0^2\left(\eta \right)cot{\delta }_0]$ (line 1) and $\mathrm{Re}\left[k{C}_0^2\left(\eta \right)(cot{\delta }_0-i)\right]$ (line 2) on energy $E$. For $E>0$ they coincide.