An evolution equation in phase space and the Weyl correspondence

AbstractThe Weyl correspondence that associates a quantum-mechanical operator to a Hamiltonian function on phase space is defined for all tempered distributions on R2. The resulting Weyl operators are shown to include most Schroedinger operators for a system with one degree of freedom. For each tempered distribution, an evolution equation in phase space is defined that is formally equivalent to the dynamics of the Heisenberg picture. The evolution equation is studied both through a separation of variables technique that expresses the evolution operator as the difference of two Weyl operators and through the geometric properties of the distribution. For real tempered distributions with compact support the evolution equation has a unique solution if and only if the Weyl equation does. The evolution operator has skew-adjoint extensions that solve the evolution equation if the distribution satisfies an orthogonal symmetry condition