Scattering from a scale invariant potential in two spatial dimensions leads to a class of novel identities involving the sinc function.

1.
R.
Jackiw
, “
Introducing scale symmetry
,”
Phys. Today
25
(
1
),
23
27
(
1972
).
2.
R. G.
Newton
,
Scattering Theory of Waves and Particles
, 2nd ed. (
Springer Verlag
,
1982
).
3.

This follows from partial wave analysis similar to the 2D case, as given in the text, except in 3D the phase shift is δl=π2ll+1ll+1+2mκ/2 and the sum l=02l+1sin2δl is then divergent because of the additional 2l + 1 multiplicity factor.

4.

If the potential is cut-off outside some large finite radius, R, such that V=κΘRr/r2 where Θ is the Heaviside step function, then the scattering amplitudes are given in terms of Bessel functions as a ratio of Wronskians, e2iδl=WJνlkR,Hl2kR/WJνlkR,Hl1kR, where νl=l2+2mκ/2, with the result (4) recovered in the limit kR → ∞.

5.
G. H.
Hardy
and
J. E.
Littlewood
, “
Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes
,”
Acta Math.
41
,
119
196
(
1916
).
6.
R.
Baillie
,
D.
Borwein
, and
J. M.
Borwein
, “
Surprising sinc sums and integrals
,”
Am. Math. Mon.
115
,
888
901
(
2008
).
You do not currently have access to this content.