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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.
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