  We present a systematic perturbative construction of the most general metric
operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians
of the standard form, H= p^2/2 + v(x), in one dimension. We show that this
problem is equivalent to solving an infinite system of iteratively decoupled
hyperbolic partial differential equations in (1+1)-dimensions. For the case
that v(x) is purely imaginary, the latter have the form of a nonhomogeneous
wave equation which admits an exact solution. We apply our general method to
obtain the most general metric operator for the imaginary cubic potential,
v(x)=i \epsilon x^3. This reveals an infinite class of previously unknown CPT-
as well as non-CPT-inner products. We compute the physical observables of the
corresponding unitary quantum system and determine the underlying classical
system. Our results for the imaginary cubic potential show that, unlike the
quantum system, the corresponding classical system is not sensitive to the
choice of the metric operator. As another application of our method we give a
complete characterization of the pseudo-Hermitian canonical quantization of a
free particle moving in real line that is consistent with the usual choice for
the quantum Hamiltonian. Finally we discuss subtleties involved with higher
dimensions and systems having a fixed length scale.
