  We study the statistics of last-passage time for linear diffusions. First we
present an elementary derivation of the Laplace transform of the probability
density of the last-passage time, thus recovering known results from the
mathematical literature. We then illustrate them on several explicit examples.
In a second step we study the spectral properties of the Schr\"{o}dinger
operator associated to such diffusions in an even potential $U(x) = U(-x)$,
unveiling the role played by the so-called Weyl coefficient. Indeed, in this
case, our approach allows us to relate the last-passage times for dual
diffusions (i.e., diffusions driven by opposite force fields) and to obtain new
explicit formulae for the mean last-passage time. We further show that, for
such even potentials, the small time $t$ expansion of the mean last-passage
time on the interval $[0,t]$ involves the Korteveg-de Vries invariants, which
are well known in the theory of Schr\"odinger operators. Finally, we apply
these results to study the emptying time of a one-dimensional box, of size $L$,
containing $N$ independent Brownian particles subjected to a constant drift. In
the scaling limit where both $N \to \infty$ and $L \to \infty$, keeping the
density $\rho = N/L$ fixed, we show that the limiting density of the emptying
time is given by a Gumbel distribution. Our analysis provides a new example of
the applications of extreme value statistics to out-of-equilibrium systems.
