Remarks on the Existence of Solutions of the Two‐Particle Lippmann‐Schwinger Equation

Abstract

We prove the following: The kernel Kl of the partial wave Lippmann‐Schwinger (LS) equation for the lth partial wave with complex energy having positive‐ or negative‐definite imaginary part belongs to the Hilbert‐Schmidt (L 2) class if the potential V is spherically symmetric and such that lim lim r→0 r δ V(r)=0,−∞<δ< 3 2 ,V(r) r→∞ =O(r −η ),η> 1 2 ,(B) for V(r)=g/r α ,1<α< 3 2 , the kernel K of the full Lippmann‐Schwinger equation satisfies Tr {(K † ) m K m }<∞, if α>(2m+1)/2m,m=2,3,⋯ .For ½ < α ≤ 1, Tr {(K †) mKm } is not absolutely convergent for any finite m, even though, for each partial wave, Kl belongs to L 2 class. An appendix deals with obtaining expressions for the T matrix in terms of eigensolutions of the mth iterated kernel when it belongs to L 2 class.