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The rigorous study of spectral stability of strong detonations was begun byErpenbeck in the 1960s. Working with the Zeldovitch-von Neumann-D\"oring (ZND)model, he identified two fundamental classes of detonation profiles, referredto as those of decreasing (D) and increasing (I) type, which appeared toexhibit very different behavior with respect to high-frequency perturbations.Using a combination of rigorous and non-rigorous arguments, Erpenbeck concludedthat type I detonations were unstable to some oscillatory perturbations forwhich the (vector) frequency was of arbitrarily large magnitude, while type Ddetonations were stable provided the frequency magnitude was sufficiently high.For type D detonations Erpenbeck's methods did not allow him to obtain a cutoffmagnitude for stability that was \emph{uniform} with respect to frequencydirection. Thus, he left open the question whether the cutoff magnitude forstability might approach $+\infty$ as certain frequency directions wereapproached. In this paper we show by quite different methods that for type Ddetonations there exists a uniform cutoff magnitude for stability independentof frequency direction. By reducing the search for unstable frequencies to abounded frequency set, the uniform cutoff obtained here is a key step towardthe rigorous validation of a number of results in the computational detonationliterature. The main difficulty in the analysis is to treat "turning points atinfinity".

High-frequency stability of multidimensional ZND detonations