Some Remarks on Effective Range Formula in Potential Scattering

Details Some Remarks on Effective Range Formula in Potential Scattering Some Remarks on Effective Range Formula in Potential Scattering

2008-12-16

arxiv.org/abs/0812.3047v1

Physics

Mathematical Physics

Scientific paper

10.1088/1751-8113/42/12/125302

In this paper, we present different proofs of very recent results on the necessary as well as sufficient conditions on the decrease of the potential at infinity for the validity of effective range formulas in 3-D in low energy potential scattering (Andr\'e Martin, private communication, to appear. See Theorem 1 below). Our proofs are based on compact formulas for the phase-shifts. The sufficiency conditions are well-known since long. But the necessity of the same conditions for potentials keeping a constant sign at large distances are new. All these conditions are established here for dimension 3 and for all angular momenta $\ell \geq 0$.

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