  A path-integral approach is used to study the spectral properties of the
generators of the SO(2,1) symmetry of conformal quantum mechanics (CQM). In
particular, we consider the CQM version that corresponds to the weak-coupling
regime of the inverse square potential. We develop a general framework to
characterize a generic symmetry generator $G$ (linear combinations of the
Hamiltonian $H$, special conformal operator $K$, and dilation operator $D$),
from which the path-integral propagators follow, leading to a complete spectral
decomposition. This is done for the three classes of operators: elliptic,
parabolic, and hyperbolic. We also highlight novel results for the hyperbolic
operators, with a continuous spectrum, and their quantum-mechanical
interpretation. The spectral technique developed for the eigensystem of
continuous-spectrum operators can be generalized to other operator problems.
