  The rigorous study of spectral stability for strong detonations was begun by
J.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-D\"oring (ZND)
model, which assumes a finite reaction rate but ignores effects like viscosity
corresponding to second order derivatives, he used a normal mode analysis to
define a stability function $V(\tau,\eps)$ whose zeros in $\Re \tau>0$
correspond to multidimensional perturbations of a steady detonation profile
that grow exponentially in time. Later in a remarkable paper [Er3] he provided
strong evidence, by a combination of formal and rigorous arguments, that for
certain classes of steady ZND profiles, unstable zeros of $V$ exist for
perturbations of sufficiently large transverse wavenumber $\eps$, even when the
von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in
the sense defined (nearly twenty years later) by Majda. In spite of a great
deal of later numerical work devoted to computing the zeros of $V(\tau,\eps)$,
the paper \cite{Er3} remains the only work we know of that presents a detailed
and convincing theoretical argument for detecting them.
  The analysis in [Er3] points the way toward, but does not constitute, a
mathematical proof that such unstable zeros exist. In this paper we identify
the mathematical issues left unresolved in [Er3] and provide proofs, together
with certain simplifications and extensions, of the main conclusions about
stability and instability of detonations contained in that paper.
  The main mathematical problem, and our principal focus here, is to determine
the precise asymptotic behavior as $\eps\to \infty$ of solutions to a linear
system of ODEs in $x$, depending on $\eps$ and a complex frequency $\tau$ as
parameters, with turning points $x_*$ on the half-line $[0,\infty)$.
