On the Ackerberg–O'Malley Resonance

Abstract:

In this paper, we continue our study of the boundary value problem

where A, B are prescribed constants and 0 < ε ≪ 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) ∼ αx and b(x) ∼ β as x → 0. In our previous work, the problem has been studied for both α > 0 and α < 0 except for the cases β/α = 1, 2, 3, . . . when α > 0 and β/α = 0,−1,−2, . . . when α < 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.