Attractive Bose-Einstein condensates in anharmonic traps: Accurate numerical treatment and the intriguing physics of the variance

•BECs with finite-range attractive interaction in anharmonic 1D traps.•Ground-state and out-of-equilibrium dynamics computed by MCTDHB.•Convergence of occupation numbers, position and momentum variance.•Density narrows, variance increases along condensation-fragmentation pathway.•100% condensation in the infinite-particle limit numerically shown. The dynamics of attractive bosons trapped in one dimensional anharmonic potentials is investigated. Particular emphasis is put on the variance of the position and momentum many-particle operators. Coupling of the center-of-mass and relative-motion degrees-of-freedom necessitates an accurate numerical treatment. The multiconfigurational time-dependent Hartree for bosons (MCTDHB) method is used, and high convergence of the energy, depletion and occupation numbers, and position and momentum variances is proven numerically. We demonstrate for the ground state and out-of-equilibrium dynamics, for condensed and fragmented condensates, for small systems and en route to the infinite-particle limit, that intriguing differences between the density and variance of an attractive Bose-Einstein condensate emerge. Implications are briefly discussed.