Finite-dimensional -symmetric Hamiltonians

Abstract

This paper investigates finite-dimensional representations of PT-symmetric\nHamiltonians. In doing so, it clarifies some of the claims made in earlier\npapers on PT-symmetric quantum mechanics. In particular, it is shown here that\nthere are two ways to extend real symmetric Hamiltonians into the complex\ndomain: (i) The usual approach is to generalize such Hamiltonians to include\ncomplex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real\nsymmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the\nfirst approach the spectrum remains real, while in the second approach the\nspectrum remains real if the PT symmetry is not broken. Both generalizations\ngive a consistent theory of quantum mechanics, but if D>2, a D-dimensional\nHermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional\nPT-symmetric matrix Hamiltonian.\n