  Using an ansatz motivated by the classical form of $e^{i\phi}$, where $\phi$
is the angle variable, we construct operators which satisfy the commutation
relations of the creation-annihilation operators for the anharmonic oscillator.
The matrix elements of these operators can be expressed in terms of {\it
entire} functions in the position complex plane. These functions provide
solutions of the Ricatti equation associated with the time-independent
Schr\"odinger equation. We relate the normalizability of the eigenstates to the
global properties of the flows of this equation. These exact results yield
approximations which complement the WKB approximation and allow an arbitrarily
precise determination of the energy levels. We give numerical results for the
first 10 levels with 30 digits. We address the question of the quantum
integrability of the system.
