A method based on the Euler-Maclaurin formula is proposed to obtain all finite-size corrections to the energies of the ground and excited states for the one-dimensional Bose gas with delta function interactions. Scaling dimensions for all gapless states and some states having a macroscopic momentum are obtained.
Thermodynamic quantities of solvable models such as one–dimensional Heisenberg model can be obtained through the finite temperature Baxter formula. For the low–temperature expansions, the estimate of the finite size corrections by use of the Euler–Maclaurin formula is essential. There has been a question on the convergence of such calculations. In this paper, choosing the XX model as a simple example, we show explicitly that the Euler–Maclaurin formula reproduces the correct low–temperature expansion of the free energy.
We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state.