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In its ground state representation, the infinite spin 1/2 Heisenberg chain provides a model for spin wave scattering that entails many features of the quantum mechanical N-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, for all numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.

Explicit Plancherel theorem for ground state representation of the Heisenberg chain