Variational Method for Scattering Length

The properties of the scattering length obtained by Kohn's method, which is one of Hulthén's variational methods, are studied by assuming a linear trial function with $n$ adjustable parameters. The scattering length $A^{\mathrm{}\left(n\right)\mathrm{}}$ decreases monotonically as the number of adjustable parameters $n$ increases, if there is no bound state in the system. This conclusion essentially comes from the upper bound theorem of Spruch and Rosenberg. When the system has $m$ bound states, the scattering length increases in value only $m$ times, and otherwise decreases monotonically. Therefore, after one verifies the presence of $m$ increases, the calculated value is certain to give an upper bound on the scattering length. The connection between the result above and the condition of Rosenberg, Spruch, and O'Malley is considered. In the Appendix comparison is made of the scattering length $A^{\mathrm{}\left(n\right)\mathrm{}}$ obtained by Hulthén's original method and Kohn's method when $m$ bound states exist in general.