  Theory of the quantum quartic oscillator is developed with close attention to
the energy cutoff one needs to impose on the system in order to approximate the
smallest eigenvalues and corresponding eigenstates of its Hamiltonian by
diagonalizing matrices of limited size. The matrices are obtained by evaluating
matrix elements of the Hamiltonian between the associated harmonic-oscillator
eigenstates and by correcting the computed matrices to compensate for their
limited dimension, using the Wilsonian renormalization-group procedure. The
cutoff dependence of the corrected matrices is found to be described by a
spiral motion of a three-dimensional vector. This behavior is shown to result
from a combination of a limit-cycle and a floating fixed-point behaviors, a
distinct feature of the foundational quantum system that warrants further
study. A brief discussion of the research directions concerning renormalization
of polynomial interactions of degree higher than four, spontaneous symmetry
breaking and coupling of more than one oscillator through the near neighbor
couplings known in condensed matter and quantum field theory, is included.
