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In this thesis, we construct an approximate series solution of the Wignerequation in terms of Airy functions, which are semiclassically concentrated oncertain Lagrangian curves in two-dimensional phase space. These curves aredefined by the eigenvalues and the Hamiltonian function of the associatedone-dimensional Schr\"odinger operator, and they play a crucial role in thequantum interference mechanism in phase space. We assume that the potential ofthe Schr\"odinger operator is a single potential well, such that the spectrumis discrete.  The construction starts from an eigenfunction series expansion of thesolution, which is derived here for first time in a systematic way, bycombining the elementary technique of separation of variables with involvedspectral results for the Moyal star exponential operator. The eigenfunctions ofthe Wigner equation are the Wigner transforms of the Schr\"odingereigenfunctions, and they are approximated in terms of Airy functions by auniform stationary phase approximation of the Wigner transforms of the WKBexpansions of the Schr\"odinger eigenfunctions.  The approximation of the eigenfunction series is an approximated solution ofthe Wigner equation, which by projection onto the configuration space providesan approximate wave amplitude, free of turning point singularities. It isgenerally expected that, the derived wave amplitude is bounded, and correctlyscaled, even on caustics, since only finite terms of the approximate terms aresignificant for WKB initial wave functions with finite energy.  The details of the calculations are presented for the simple potential of theharmonic oscillator, in order to be able to check our approximationsanalytically. But, the same construction can be applied to any potential well,which behaves like the harmonic oscillator near the bottom of the well.

Construction of an approximate solution of the Wigner equation by uniformization of WKB functions