  This paper extends and clarifies results of Steinsaltz and Evans [Trans.
Amer. Math. Soc. 359 (2007) 1285-1234], which found conditions for convergence
of a killed one-dimensional diffusion conditioned on survival, to a
quasistationary distribution whose density is given by the principal
eigenfunction of the generator. Under the assumption that the limit of the
killing at infinity differs from the principal eigenvalue we prove that
convergence to quasistationarity occurs if and only if the principal
eigenfunction is integrable. When the killing at infinity is larger than the
principal eigenvalue, then the eigenfunction is always integrable. When the
killing at infinity is smaller, the eigenfunction is integrable only when the
unkilled process is recurrent; otherwise, the process conditioned on survival
converges to 0 density on any bounded interval.
