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The forward BFKL equation is discretised in virtuality space and it is shownthat the diffusion into infrared and ultraviolet momenta can be understood interms of a semi-infinite matrix. The square truncation of this matrix can beexponentiated leading to asymptotic eigenstates sharing many features with theBFKL gluon Green's function in the limit of large matrix size. This truncationis closely related to a representation of the XXX Heisenberg spin $= -\frac{1}{2}$ chain with SL(2) invariance where the Hamiltonian acts on asymmetric double copy of the harmonic oscillator. A simple modification of theBFKL matrix suppressing the infrared modes generates evolution with energycompatible with unitarity.

A semi-infinite matrix analysis of the BFKL equation