  We investigate symmetric oscillators, and in particular their quantization,
by employing semiclassical and quantum phase functions introduced in the
context of Liouville-Green transformations of the Schr\"{o}dinger equation. For
anharmonic oscillators, first order semiclassical quantization is seldom
accurate and the higher order expansions eventually break down given the
asymptotic nature of the series. A quantum phase that allows in principle to
retrieve the exact quantum mechanical quantization condition and wavefunctions
is given along with an iterative scheme to compute it. The arbitrariness
surrounding quantum phase functions is lifted by supplementing the phase with
boundary conditions involving high order semiclassical expansions. This allows
to extend the definition of oscillation numbers, that determine the
quantization of the harmonic oscillator, to the anharmonic case. Several
illustrations involving homogeneous as well as coupling constant dependant
anharmonic oscillators are given.
