LOW-ENERGY EXPANSION OF S(K) MATRIX FOR SCHRODINGER EQUATION WITH A CERTAIN TYPE OF SPHERICALLY SYMMETRIC POTENTIALS

The potential V(γ) treated is generally of such a long-range type that, when terms which for large γ decrease at least exponentially are neglected, the limit potential equals {Mathematical expression} where e n>0 for all n. The S L(k) matrix corresponding to the L-th partial wave can in this case be reached in two steps. First the phase shifts produced by the above-defined limit potential V 1(r) are determined. Subsequently the effect of the difference potential V 1(r)=V(r)-V 1(r) can be determined. The S L(k) matrix is now obtained in a form, which is convergent (1) when |k|<p. Here p is the largest value which makes the integral {Mathematical expression}. convergent. The case (2-4)V(r) =cr -1 +dr -2 +V e (r), which is of obvious importance, will be studied explicitly. In the case that {Mathematical expression} the physical solution takes for large r the form u L Raa (k, r) ∼ sin (kr -Lπ/2 -γ log 2 kr +δ L (k) +η L (k)). Here δ L (k) is the phase shift produced by the difference potential V e(r) while η L (k) = arg Γ(Γ′ + 1 +iγ) + (L -L′)π/2 is the phase shift produced by the limit potential V 1(r). We obtain {Mathematical expression} where γ=c/2 k and L′=(1/4+L(L+1)=d)1/2-1/2 differ from an integer or half-integer. Here a L n and b L n are independent of k and essentially integrals over all space and whose integrands decrease at least exponentially for large values of the arguments, a L n and b L n are therefore well suited for numerical calculations. © 1968 Società Italiana di Fisica.