Minimum Property in the Hulthén-Type Variational Methods

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Abstract
The nature of the extremum of the phase shift in the variation principles, which are all based on the Schrödinger differential equation directly, is investigated in detail. If the system has no bound state, the scattering length determined by the original Hulthén method (5, 29) is proved to have the minimum character. The minimum nature is maintained even for the system which has several bound states, if the trial function is taken to be orthogonal to the bound state wave functions. The Kohn method and the second Hulthén method (15) give neither an upper bound nor a lower bound for the phase shift in general. The sufficient condition is, however, obtained for each case, under which the stationary expression for the scattering length has the minimum character. The condition does not hold true for the electron-hydrogen atom scattering for the Kohn method and for the second Hulthén method. It is pointed out that a comparison of approximate values obtained by different methods using the same trial function does not afford any information about the ``proper source of errors'' of the results.

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