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

TL;DR

This paper investigates the conditions under which block-diagonalizing transformations exist for semiclassical ODEs, highlighting differences between analytic and smooth coefficients and addressing longstanding questions in the field.

Contribution

It establishes the existence of such transformations near infinity for analytic ODEs and demonstrates the limitations near finite points, clarifying the role of coefficient regularity.

Findings

Bounded transformations exist only locally near finite points.

Analytic coefficients allow for transformations under spectral separation.

Counterexamples show sharpness of hypotheses for existence of transformations.

Abstract

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 $C^\omega$ vs. $C^\infty$ stationary phase estimates