By K. Osterwalder

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**Extra resources for Mathematical Problems in Theoretical Physics**

**Sample text**

120). If s ≤ 2 + 3 the integral diverges and the first term in the effective range expansion is no longer defined. 138) only converges for large r if s > 2 + 5 and consequently if s ≤ 2 + 5 the second term in the effective range expansion is not defined. 139) and so on for higher terms in the effective range expansion. An important example of long-range potentials occurs in elastic electron scattering by an atom in a non-degenerate s-wave ground state such as atomic hydrogen or the inert gases.

The radial Schrödinger equation then has the asymptotic form ( + 1) A d2 − − 2 + k 2 u (r ) = 0, 2 2 dr r r r ≥a. 150) which has the solution λ = − 12 ± 1 2 (2 + 1)2 + 4A 1/2 . 149) reduces to the standard form d2 λ(λ + 1) − + k 2 u (r ) = 0, dr 2 r2 r ≥a. 152) where λ is in general a non-integral quantity. 151) so that λ → in the limit A → 0. 155) which defines the K -matrix K λ (k). 155). 156) where τ = 12 π( − λ) . 157) It follows that when A = 0 then = λ and K (k) = K λ (k). 118) by relating K λ (k) to the R-matrix on the boundary r = a.

120) that the low-energy s-wave cross section σ0 = 4πa02 1 4π 4π 2 = sin δ (k) = . 127) The zero-energy cross section is thus 4πa02 . Also, when an s-wave bound state occurs at zero energy then the scattering length and hence the cross section is infinite. We now determine the behaviour of the cross section when an s-wave bound state occurs close to zero energy. 26) that T (k) = S (k) − 1 = 2i . 128) Hence a pole in the S- and T -matrices occurs when cot δ (k) = i. However, we saw in Sect. 3, see Fig.