Gradient corrections to the local-density approximation in the one-dimensional Bose gas

The local-density approximation (LDA) is the central technical tool in modeling of quantum gases trapping potentials. It consists treating gas as an assembly independent mesoscopic fluid cells at equilibrium with a local chemical potential, and it justified when correlation length larger than size cells. LDA often regarded crude approximation, particularly ground state one-dimensional (1D) Bose gas, where ``therefore said to be'' infinite (in sense that functions decay power law). Here we take another look LDA. density $\ensuremath{\rho}(x)$ viewed functional potential $V(x)$, which one applies gradient expansion. zeroth order expansion first-order correction vanishes due reflection symmetry. At second order, there are two corrections proportional ${d}^{2}V/d{x}^{2}$ ${(dV/dx)}^{2}$, propose method determine corresponding coefficients by perturbative calculation Lieb-Liniger model. This leads expression for terms matrix elements operator, can principle be evaluated numerically arbitrary coupling constant; here show how efficiently evaluate coefficient associated curvature ${d}^{2}V/d{x}^{2}$, dominates deviation near minima or maxima potential. Both analytically limits repulsion (hard-core bosons) small (quasicondensate). corrected profiles compared density-matrix renormalization group calculations, significant improvement zeroth-order