Finite-dimensional   -symmetric Hamiltonians

This paper investigates finite-dimensional -symmetric Hamiltonians. In doing so, it clarifies some of the claims made in earlier papers on -symmetric quantum mechanics. In particular, it is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) the usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex -symmetric Hamiltonians. In the first approach, the spectrum remains real, while in the second approach the spectrum remains real if the symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D > 2, a D -dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D -dimensional -symmetric matrix Hamiltonian.