  This article studies the quasi-stationary behaviour of absorbed
one-dimensional diffusion processes with killing on $[0,\infty)$. We obtain
criteria for the exponential convergence to a unique quasi-stationary
distribution in total variation, uniformly with respect to the initial
distribution. Our approach is based on probabilistic and coupling methods,
contrary to the classical approach based on spectral theory results. Our
general criteria apply in the case where $\infty$ is entrance and 0 either
regular or exit, and are proved to be satisfied under several explicit
assumptions expressed only in terms of the speed and killing measures. We also
obtain exponential ergodicity results on the $Q$-process. We provide several
examples and extensions, including diffusions with singular speed and killing
measures, general models of population dynamics, drifted Brownian motions and
some one-dimensional processes with jumps.
