Variational Upper and Lower Bounds on Phase Shifts

抄録

A variational method is developed for the calculation of upper and lower bounds on the phase shifts for the scattering of a particle by a compound system. It is essentially the combination of a generalization of Kato’s method with a couple of variational principles, i.e. Kato-Temple’s and Kohn’s methods. First, the central field version of the theory is applied to the s-wave scattering of an electron by the static potential of a hydrogen atom. A trial function with 10 parameters gives the results accurate to about 10⁻⁷ radians at the incident energies lower than 0.5 atomic units. Then rigorous bounds on the phase shifts for the s-wave elastic scattering of an electron by a hydrogen atom are calculated with a 52-parameter-trial function. The lower bounds are fairly close to other reliable published data. On the other hand, the upper bounds are not very close to them, and the difference between the both bounds for the wave number k=0.4 a.u. is 0.029 radians for the singlet scattering, and 0.18 radians for the triplet scattering.