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

The fractal geometry and the mapping of Efimov states to Bloch states