Scattering length of an attractive square-well potential
                                               7.6.2000 B. Friman

α[k_, v_] := (Sqrt[2*(e[k] + v)])

e[k_] := k^2/2

Ratio  χ'/χ  at radius of potential

γ0[v_, a_] := α[0, v]*
        Cos[α[0, v]*a]/Sin[α[0, v]*a]

Scattering length

sl[v_, a_] := a - 1/γ0[v, a]

Plot[sl[v, 2], {v,  .001, 3}]

The singularities in the scattering length correspond to the appearance of bound states.

Check that the first singularity sits at π^2/32

sl[Pi^2/32 +  .0000001, 2]

sl[Pi^2/32 -  .0000001, 2]