Anharmonic Oscillator Spectrum via Diagonalization of Amplitudes

The energy spectrum of a quantum system can be accurately calculated by the numerical diagonalization of the space-discretized matrix of its evolution operator, that is, the matrix of its transition amplitudes. Here we calculate the spectrum of a one-dimensional 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.