Scattering on a modified Pöschl-Teller potential

The potential (1.89) considered in the previous section might be inconvenient in some cases, as it produces large gradients of the solution at its border $r\approx R$ due to the abrupt change of its value from $-V_0$ to zero. This can be avoided by using, for example, the modified Poschl-Teller potential

$$V(r) = -\frac{V_0}{\mathop{\rm ch}\nolimits ^2 (r/R)} =-\frac{\hbar^2}{2mR^2}\frac{\lambda(\lambda-1)}{\mathop{\rm ch}\nolimits ^2 (r/R)},$$ (1.97)

where $V_0$ is the depth of the potential and $R$ is its range.

The problem of three-dimensional scattering on this potential can be solved analytically (see, e.g. [Flu71]) and the dependence of the $s$-wave scattering length on the depth of the potential well can be found explicitly:

$$\frac{a_{3D}}{R}=\frac{\pi}{2}\mathop{\rm ctg}\nolimits \frac{\pi\lambda}{2}+\gamma+\Psi(\lambda),$$ (1.98)

where $\gamma = 0.5772...$ is the Euler's constant and $\Psi$ is the Digamma function. This dependence is expressed in the Fig. 4.2.