A density matrix renormalization group approach to quantum point contacts
Using the density matrix renormalization group (DMRG) combined with the correction-vector method, we investigate the competition between an harmonic potential and repulsive interactions in a one-dimensional fermionic system. The parabolic confinement induces spatial inhomogeneity, and by tuning its curvature one can continuously interpolate between a potential well--relevant for cold-atom setups--and a quantum barrier, as realized in mesoscopic systems such as quantum point contacts. We analyze how the ground-state particle distribution evolves with the strength and sign of the confining potential and how the confinement reshapes the spectral weight of the local density of states (LDOS) at the center of the chain. In the barrier regime, a localized peak emerges in the electron part of the spectrum ($ω>0$) as a direct consequence of the potential. In contrast, in the well configuration and for weak interactions, a localized feature persists but shifts to the hole sector ($ω<0$). However, for stronger interactions, the LDOS no longer displays clear signatures of the external potential, indicating that correlations dominate over single-particle confinement.