Some Remarks on Effective Range Formula in Potential Scattering

TL;DR

This paper investigates the conditions under which effective range formulas are valid in 3-D low energy potential scattering, providing new proofs for the necessity of potential decay conditions for potentials with constant sign.

Contribution

It offers new proofs of the necessary conditions for the validity of effective range formulas, extending results to all angular momenta in three dimensions.

Findings

Established necessary and sufficient decay conditions for potentials with constant sign.

Extended validity of effective range formulas to all angular momenta in 3-D.

Provided new proofs based on compact phase-shift formulas.

Abstract

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 $\ell \geq 0$.