  We show that the eigenvectors of the PT-symmetric imaginary cubic oscillator
are complete, but do not form a Riesz basis. This results in the existence of a
bounded metric operator having intrinsic singularity reflected in the
inevitable unboundedness of the inverse. Moreover, the existence of non-trivial
pseudospectrum is observed. In other words, there is no quantum-mechanical
Hamiltonian associated with it via bounded and boundedly invertible similarity
transformations. These results open new directions in physical interpretation
of PT-symmetric models with intrinsically singular metric, since their
properties are essentially different with respect to self-adjoint Hamiltonians,
for instance, due to spectral instabilities.
