A common approach to studying many body quantum mechanics is to start with a related simpler problem that can be solved exactly and then to take into account corrections as a perturbation. This seems to be particularly viable for one-dimensional (1D) systems, where there exists a large class of exactly solvable models. The first steps for developing a perturbation theory from these models requires knowing the energy eigenvalues as well as the corresponding eigenstates, which need to be properly normalized.
In this talk I will discuss the norms of the Bethe states, a certain basis of stationary states, for the Heisenberg spin-1/2 chain. The analysis is based on the so-called ODE/IQFT correspondence, which, together with numerical work has allowed us to formulate a series of conjectures for their form in the limit when the size of the spin chain goes to infinity. The talk will be based on the recent paper with my collaborator S. L. Lukyanov: arXiv:1906.07081.