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Block-diagonalization of ODEs in the semiclassical limit and $C^ω$ vs. $C^\infty$ stationary phase

Motivated by issues in detonation stability, we study existence of block-diagonalizing transformations for ordinary differential semiclassical limit problems arising in the study of high-frequency eigenvalue problems. Our main results are to (i) establish existence of block-diagonalizing transformations in a neighborhood of infinity for analytic-coefficient ODE, and (ii) establish by a series of counterexample sharpness of hypotheses and conclusions on existence of block-diagonalizing transformations near a finite point. In particular, we show that, in general, bounded transformations exist only locally, answering a question posed by Wasow in the 1980's, and, under the minimal condition of spectral separation, for ODE with analytic rather than $C^\infty$ coefficients. The latter issue is connected with quantitative comparisons of $C^ω$ vs. $C^\infty$ stationary phase estimates