  Efimov states are known to have a discrete real space scale invariance,
working in momentum space we identify the relevant discrete scale invariance
for the scattering amplitude defining its Weierstrass function as well. Through
the use of the mathematical formalism for discrete scale invariance for the
scattering amplitude we identify the scaling parameters from the pole structure
of the corresponding zeta function, it's zeroth order pole is fixed by the
Efimov physics. The corresponding geometrical fractal structure for Efimov
physics in momentum space is identified as a ray across a logarithmic spiral.
This geometrical structure also appears in the physics of atomic collapse in
the relativistic regime connecting it to Efimov physics. Transforming to
logarithmic variables in momentum space we map the three-body scattering
amplitude into Bloch states and the ladder of energies of the Efimov states are
simply obtained interms of the Bohr-Sommerfeld quantization rule. Thus through
the mapping the complex problem of three-body short range interaction is
transformed to that of a non-interacting single particle in a discrete lattice.

Nuclear Theory;Quantum Gases