Block-diagonalization of ODEs in the semiclassical limit and \(C^\omega\) versus \(C^\infty\) stationary phase (Q2806197)

Motivated by problems in detonation and related hydrodynamical and continuum-mechanical stability, this paper considers the general semiclassical limit problem NEWLINE\[NEWLINEh\frac{dZ}{dx} = (A(x,h;q) + h B(x,h;q))Z \qquad Z\in {\mathbb C}^N, \;\; h \to 0^+ \tag{1}NEWLINE\]NEWLINE on a possibly unbounded domain \(x \in [a, b] \in {\mathbb R},\) representing a generalized spectral problem with wavelength \(h\in{\mathbb R}^+\) and frequency \(k = \frac{1}{h}\). Here, \(q\in {\mathbb R}^s\), bounded, records any additional parameters associated with the problem: typically, spectral angle and/or bifurcation parameters.NEWLINENEWLINEThe goal of this paper is a systematic treatment of local block reduction of (1), or decomposition of the equations into spectrally separated blocks possessing nontrivial turning points, in particular in the important case of a neighborhood of plus or minus infinity.NEWLINENEWLINEThe main results are: {\parindent=6mm \begin{itemize}\item[(i)] the proof of the existence of block-diagonalizing transformations in a neighborhood of infinity for analytic-coefficient ordinary differential equations (ODEs), \item[(ii)] by a series of counterexamples to emphasize the sharpness of the assumptions and conclusions on the existence of block-diagonalizing transformations near a finite point.NEWLINENEWLINE\end{itemize}} In particular, it is shown that, in general, bounded transformations exist only locally, answering a question posed by \textit{W. Wasow} [Linear turning point theory. New York etc.: Springer-Verlag (1985; Zbl 0558.34049)], and, under the minimal condition of spectral separation, for ODEs with analytic rather than \({\mathbb C}^{\infty}\) coefficients. The latter issue is connected with quantitative comparisons of \({\mathbb C}^{\omega}\) versus \({\mathbb C}^{\infty}\) stationary phase estimates.