Exact Results in Quantum Theory

Local density approximation is a standard technique of studying trapped atomic gases. In this talk I will explore its limitations by considering the 1d Bose gas. The 1d Bose gas, modelled by the Lieb-Liniger Hamiltonian is, in the absence of the trap, exactly solvable. We will use this exact solution to build up a perturbative expansion in the trapping potential. This is a novel approach to this problem, as the perturbed model is already a strongly interacting theory and computation of the perturbation series relies on knowledge of multi-point correlation functions. Recent advances in thermodynamic form-factors and generalized hydrodynamics provides tools to study such correlators in the relevant regime. With this approach we find an intuitive picture of the local density approximation and a precise estimation of its error.