  The cutoff phenomenon describes a case where a Markov chain exhibits a sharp
transition in its convergence to stationarity. In 1996, Diaconis surveyed this
phenomenon, and asked how one could recognize its occurrence in families of
finite ergodic Markov chains. In 2004, the third author noted that a necessary
condition for cutoff in a family of reversible chains is that the product of
the mixing-time and spectral-gap tends to infinity, and conjectured that in
many settings, this condition should also be sufficient. Diaconis and
Saloff-Coste (2006) verified this conjecture for continuous-time
birth-and-death chains, started at an endpoint, with convergence measured in
separation. It is natural to ask whether the conjecture holds for these chains
in the more widely used total-variation distance.
  In this work, we confirm the above conjecture for all continuous-time or lazy
discrete-time birth-and-death chains, with convergence measured via
total-variation distance. Namely, if the product of the mixing-time and
spectral-gap tends to infinity, the chains exhibit cutoff at the maximal
hitting time of the stationary distribution median, with a window of at most
the geometric mean between the relaxation-time and mixing-time.
  In addition, we show that for any lazy (or continuous-time) birth-and-death
chain with stationary distribution $\pi$, the separation $1 - p^t(x,y)/\pi(y)$
is maximized when $x,y$ are the endpoints. Together with the above results,
this implies that total-variation cutoff is equivalent to separation cutoff in
any family of such chains.
