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The rigorous study of spectral stability for strong detonations was begun byJ.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-D\"oring (ZND)model, which assumes a finite reaction rate but ignores effects like viscositycorresponding to second order derivatives, he used a normal mode analysis todefine a stability function $V(\tau,\eps)$ whose zeros in $\Re \tau>0$correspond to multidimensional perturbations of a steady detonation profilethat grow exponentially in time. Later in a remarkable paper [Er3] he providedstrong evidence, by a combination of formal and rigorous arguments, that forcertain classes of steady ZND profiles, unstable zeros of $V$ exist forperturbations of sufficiently large transverse wavenumber $\eps$, even when thevon Neumann shock, regarded as a gas dynamical shock, is uniformly stable inthe sense defined (nearly twenty years later) by Majda. In spite of a greatdeal 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 detailedand convincing theoretical argument for detecting them.  The analysis in [Er3] points the way toward, but does not constitute, amathematical proof that such unstable zeros exist. In this paper we identifythe mathematical issues left unresolved in [Er3] and provide proofs, togetherwith certain simplifications and extensions, of the main conclusions aboutstability and instability of detonations contained in that paper.  The main mathematical problem, and our principal focus here, is to determinethe precise asymptotic behavior as $\eps\to \infty$ of solutions to a linearsystem of ODEs in $x$, depending on $\eps$ and a complex frequency $\tau$ asparameters, with turning points $x_*$ on the half-line $[0,\infty)$.

The Erpenbeck high frequency instability theorem for ZND detonations