The energy spectrum of a quantum system can be accurately calculated by the numerical diagonalization of the spacediscretized matrix of its evolution operator, that is, the matrix of its transition amplitudes. Here we calculate the spectrum of a onedimensional anharmonic oscillator with the potential , using level effective action. For a general quantum system described by the Hamiltonian , the probability for a transition from an initial state to a final state in time is calculated as , with the transition amplitude . In a recently developed effective action approach, the amplitude is expressed in terms of the effective potential. Then a set of recursive relations allows systematic analytic derivation of terms in the expansion of the effective potential in the time . The effective action thus obtained is characterized by a chosen level corresponding to the maximal order occurring in its expansion.
The effective action approach was originally developed in a series of papers: The extension of the approach to manybody systems is presented in the following papers: This method has been successfully applied to numerical studies of properties of various quantum systems: Recently, this approach has been extended to systems in timedependent potentials: Mathematica programs developed for symbolic derivation of higherorder effective actions, as well as the C programs developed and used in numerical simulations in the above papers, can be found at http://www.scl.rs/speedup.
