Low-energy expansion of theS(k) matrix for the Schrödinger equation with a certain type of spherically symmetric potentials

Abstract

The potentialV(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\(V_1 (r) = \sum\limits_{n = 1}^M {c_n r^{ - e_n } } \) whereen>0 for alln. TheSL(k) matrix corresponding to theL-th partial wave can in this case be reached in two steps. First the phase shifts produced by the above-defined limit potentialV1(r) are determined. Subsequently the effect of the difference potentialV1(r)=V(r)−V1(r) can be determined. TheSL(k) matrix is now obtained in a form, which is convergent (1) when |k| 0}^\infty {\left| {V_e (r)} \right|} \). convergent. The case (2–4)V(r) =cr−1 +dr−2 +Ve(r), which is of obvious importance, will be studied explicitly. In the case that\(\mathop {\lim }\limits_{r \to 0} r^2 V_e (r) \geqslant - d\) the physical solution takes for larger the formuLRaa (k, r) ∼ sin (kr −Lπ/2 −γ log 2kr +δL(k) +ηL(k)). HereδL(k) is the phase shift produced by the difference potentialVe(r) whileηL(k) = argΓ(Γ′ + 1 +iγ) + (L −L′)π/2 is the phase shift produced by the limit potentialV1(r). We obtain $$\begin{gathered} ctg \delta _L (k) = \frac{{1 - exp[2\gamma \pi ]}}{{\sin (2L'\pi )}} - tg(L'\pi ) + \hfill \\ + \frac{{k^{ - 2L' - 1} \Gamma (2L' + 2)\Gamma (2L' + 1)}}{{\left( {2^{L'} \left| {\Gamma (L' + 1 + i\gamma )} \right|} \right)^2 }}exp[\gamma \pi ] \left( {\sum\limits_{n = 0}^\infty {a_{L_n } k^{2n} } } \right)/\left( {\sum\limits_{n = 0}^\infty {b_{L_n } k^{2n} } } \right) \hfill \\ \end{gathered} $$ where γ=c/2k andL′=(1/4+L(L+1)=d)1/2−1/2 differ from an integer or half-integer. HereaLn andbLn are independent ofk and essentially integrals over all space and whose integrands decrease at least exponentially for large values of the arguments,aLn andbLn are therefore well suited for numerical calculations.