A canonical small divisor problem for the Nash-Moser method

Abstract

In this note we prove a general elementary small divisor theorem for Hs norms of N ×N matrices that provides a potentially useful estimate for expunging resonances in Nash-Moser Newton Iterations. The theorem requires compatibility conditions on the approximating matrices, and we investigate how the theorem can fail when the compatibility conditions are violated. This investigation suggests that establishing compatibility of the approximations, not the presence of small eigenvalues, is the main obstacle in obtaining small divisor theorems sufficient for expunging resonances in Nash-Moser Newton Iterations.