The WKB method is an important analytic tool for solving numerous problems in
mathematical physics of 1D systems, for example the stationary (time-independent)
Schrödinger equation in one dimension, or the classical dynamics of one-dimensional
time-dependent (nonautonomous) Hamilton oscillators. I shall review the standard
WKB method including the exact explicit solutions to all orders, published by Rob-
nik and Romanovski (2000), and applied in a series of papers. Among other results
we have shown that the application of the method in cases of the Schrödinger equa-
tion with exactly solvable potentials leads to an infinite series to all orders, that the
series converges and the sum reproduces the known exact eigenenergies. We shall
look in particular at the case of the time-dependent one-dimensional linear Hamilto-
nian oscillator, and then I shall present the approach towards generalizing the WKB
method for the case of one-dimensional time-dependent nonlinear Hamiltonian oscil-
lators having quadratic kinetic energy and homogeneous power law potential, which
includes e.g. the quartic oscillator, and of course also the linear oscillator. I will
show that the nonlinear method, although only in the leading approximation, is very
useful and accurate. We also shall touch upon possible generalizations.

* Marko Robnik, CAMTP, Univerza v Mariboru