The Euclidean $\phi^4_2$ theory as a limit of an inhomogeneous Bose gas

This research demonstrates that the grand canonical Gibbs state of an interacting two-dimensional quantum Bose gas, when confined by a trapping potential, converges to the complex Euclidean field theory with local quartic self-interaction under conditions of high gas density and small interaction range. The study achieves convergence of the relative partition function and in L^1 \cap L^\infty of the renormalized reduced density matrices. A key finding is that the field theory operates on distributions of negative regularity, necessitating renormalization by divergent mass and energy counterterms, which, unlike homogeneous settings, are diverging functions rather than finite scalars. This presents significant mathematical challenges and the paper also provides quantitative bounds on the Green function of Schrödinger operators, which could have independent applications.