Abstract 
The Wigner equation is a nonlocal, evolution equation in phasespace. It describes the evolution of the Weyl symbol of the density operator which, in general, is governed by the Liouvillevon Neumann equation of quantum mechanics. For pure quantum states, the Wigner equation is an equivalent reformulation of the standard quantummechanical Schrodinger equation, and it could be also derived in an operational way by considering the Wigner transform of the quantum wave function, without using the Weyl calculus.
In this thesis, we construct an approximate solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in twodimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated onedimensional Schrodinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrodinger operator is a singlewell potential such that the spectrum is dis¬crete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrodinger eigenfunctions, and they are approximated in terms of Airy functions by a uni¬form stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrodinger eigenfunctions. Although the WKB approximations of Schrodinger eigenfunctions have nonphysical singularities at the turning points of the classical Hamiltonian, the phase space eigenfunctions provide bounded, and correctly scaled, wave amplitudes when they are projected back onto the configuration space (uniformization).
Therefore, the approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approx¬imate wave amplitude, free of turning point singularities. It is generally expected that, the
derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy.
The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same con¬struction can be applied to any potential well which behaves like the harmonic oscillator near the bottom of the well. In principle, this construction could be extended to higher di¬mensions using canonical forms of the Hamiltonian functions and employing the symplectic covariance inherited by the Weyl representation into the Wigner equation.
