LAUSR

We provide the eigenfunctions for a quantum chain of N conformal spins with nearest-neighbor interaction and open boundary conditions in the irreducible representation of SO(1,5) of scaling dimension Δ=2-iλ and spin numbers ℓ=ℓ[over ˙]=0. The spectrum of the model is separated into N equal contributions, each dependent on a quantum number Y_{a}=[ν_{a},n_{a}] which labels a representation of the principal series. The eigenfunctions are orthogonal and we computed the spectral measure by means of a new star-triangle identity. Any portion of a conformal Feynmann diagram with square lattice topology can be represented in terms of separated variables, and we reproduce the all-loop "fishnet" integrals computed by B. Basso and L. Dixon via bootstrap techniques. We conjecture that the proposed eigenfunctions form a complete set and provide a tool for the direct computation of conformal data in the fishnet limit of the supersymmetric N=4 Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.