ABSTRACT

This chapter, as well as the preceding one, deals with the problem of constructing the high-frequency asymptotics of the Green function for stationary equations containing a large parameter k. In Chap­ ter 6, we showed, using the Maxwell equations as an example, how equations whose Hamilto­ nians do not have finite motions can be treated. Stating this condition in a different form, we can say that these Hamiltonians possess the nontrapping property: the trajectories of the Hamilto­ nian vector field on the zero level of the Hamiltonian exit to infinity in the rr-direction, i.e., leave π “ 1 (K ) Є in finite time for an arbitrary compact set K C W į. Here π : T*Wį -» R£ is the natural projection. In this chapter we consider Hamiltonians that still satisfy the condition

but may be trapping (i.e., the Hamiltonian vector field may have finite motions). In general, the construction outlined in Section 6.3-6.4 fails for such equations and hardly can

be recovered (for example, it is not clear what to do if V (H ) is ergodic on some compact component of char # . However, for ordinary differential equations the situation is much simpler, since finite motions under the condition that the Hamiltonian vector field has no fixed points on the character­ istic variety are closed trajectories, and the construction of the Green function can be appropriately modified to allow considering closed trajectories. Thus, in Sections 7.1-7.4 we consider the general case of ordinary differential equations satisfying this condition, and in Section 7.5 an example is given pertaining to short electromagnetic wave propagation in ionosphere. (Although this problem is three-dimensional, we assume harmonic dependence of the solution on two of the three coor­ dinates, which in effect reduces the problem to an ordinary differential equation.) The main new feature occurring in the presence of closed trajectories is the resonance phenomenon (the diver­ gence of the asymptotic solution for some values of the wave number k), which leads to physically meaningful effects in specific problems.