On the Ackerberg-O'Malley resonance

摘要

In this paper, we continue our study of the boundary value problem [GRAPHICS] where A, B are prescribed constants and 0 < epsilon 1 is a small positive parameter. We assume that the coefficients a(x) and b(x) are sufficiently smooth functions with the behavior given by a(x) similar to a(x) and b(x) similar to beta as x --> 0. In our previous work, the problem has been studied for both alpha > 0 and alpha < 0 except for the cases beta/alpha = 1, 2, 3.... when alpha > 0 and beta/alpha = 0, -1, -2,... when a < 0. In the present paper, we study these exceptional cases and obtain, by rigorous analysis, uniformly valid asymptotic solutions of the problem. From these solutions, we also show that the conditions in these exceptional cases are exactly the ones which are necessary and sufficient for the Ackerberg-O'Malley resonance. [References: 10]