We investigate symmetric oscillators, and in particular their quantization,by employing semiclassical and quantum phase functions introduced in thecontext of Liouville-Green transformations of the Schr\"{o}dinger equation. Foranharmonic oscillators, first order semiclassical quantization is seldomaccurate and the higher order expansions eventually break down given theasymptotic nature of the series. A quantum phase that allows in principle toretrieve the exact quantum mechanical quantization condition and wavefunctionsis given along with an iterative scheme to compute it. The arbitrarinesssurrounding quantum phase functions is lifted by supplementing the phase withboundary conditions involving high order semiclassical expansions. This allowsto extend the definition of oscillation numbers, that determine thequantization of the harmonic oscillator, to the anharmonic case. Severalillustrations involving homogeneous as well as coupling constant dependantanharmonic oscillators are given.
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