  The generating functionals for the local composite operators, $\Phi^2(x)$ and
$\Phi^4(x)$, are used to study excitations in the scalar quantum field theory
with $\lambda \Phi^4$ interaction. The effective action for the composite
operators is obtained as a series in the Planck constant $\hbar$, and the two-
and four-particle propagators are derived. The numerical results are studied in
the space-time of one dimension, when the theory is equivalent to the quantum
mechanics of an anharmonic oscillator. The effective potential and the poles of
the composite propagators are obtained as series in $\hbar$, with effective
mass and coupling determined by non-perturbative gap equations. This provides a
systematic approximation method for the ground state energy, and for the second
and fourth excitations. The results show quick convergence to the exact values,
better than that obtained without including the operator $\Phi^4$.
