  We present a global linear stability analysis of nuclear fuel accumulating on
the surface of an accreting neutron star and we identify the conditions under
which thermonuclear bursts are triggered. The analysis reproduces all the
recognized regimes of hydrogen and helium bursts, and in addition shows that at
high accretion rates, near the limit of stable burning, there is a regime of
``delayed mixed bursts'' which is distinct from the more usual ``prompt mixed
bursts.'' In delayed mixed bursts, a large fraction of the fuel is burned
stably before the burst is triggered. Bursts thus have longer recurrence times,
but at the same time have somewhat smaller fluences. Therefore, the parameter
alpha, which measures the ratio of the energy released via accretion to that
generated through nuclear reactions in the burst, is up to an order of
magnitude larger than for prompt bursts. This increase in alpha near the
threshold of stable burning has been seen in observations. We explore a wide
range of mass accretion rates, neutron star radii and core temperatures, and
calculate a variety of burst properties. From a preliminary comparison with
data, we suggest that bursting neutron stars may have hot cores, with T_{core}
>~ 10^{7.5} K, consistent with interior cooling via the modified URCA or
similar low-efficiency process, rather than T_{core} ~ 10^7 K, as expected for
the direct URCA process. There is also an indication that neutron star radii
are somewhat small <~ 10 km. Both of these conclusions need to be confirmed by
comparing more careful calculations with better data.
