Completeness and orthonormality in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>PT</mml:mi></mml:math>-symmetric quantum systems

Some $\mathrm{PT}$-symmetric non-Hermitian Hamiltonians have only real eigenvalues. There is numerical evidence that the associated $\mathrm{PT}$-invariant energy eigenstates satisfy an unconventional completeness relation. An ad hoc scalar product among the states is positive definite only if a recently introduced ``charge operator'' is included in its definition. A simple derivation of the conjectured completeness and orthonormality relations is given. It exploits the fact that $\mathrm{PT}$ symmetry provides a link between the eigenstates of the Hamiltonian and those of its adjoint, forming a dual pair of bases. The charge operator emerges naturally upon expressing the properties of the dual bases in terms of one basis only, and it is shown to be a function of the Hamiltonian.