Projection Operator Method for Collective Tunneling Transitions

The evolution of a state in a many-body system can be formulated in either Schrodinger picture or the Heisenberg picture.The evolution of the Heisenberg operator for a density matrix is given by the Liouville equation.The Liouville evolu- tion function has an analyticity that differs from the Schrodinger evolution function, because the Liouville eigenvalue equation yields the splitting of two energy eigenval- ues, while Schrodinger equation yields each of the energy eigenvalues.However, no method to exactly solve the Liouville evolution equation for a given Hamiltonian has yet been established.With regard to the splitting of two nearly degenerate energy eigenvalues in the context of collective tunneling transitions, we obtain an expres- sion for the energy splitting from the analytical structure of the spectral function for the Liouville evolution employing the projection operator method.The projection operator method for the Liouville equation is powerful for the purpose of analyzing the evolution of a density matrix. The projection operator method for the Liouville equation has been extensively applied to analyze the evolution of many-body systems. 1) The evolution of a system expressed in terms of time convolution is usually formulated to second order in the perturbation Hamiltonian. 2) When the second order calculation does not yield a rea- sonable result, however, it is necessary to develop an exact expression.In particular,