PTPT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues

The mathematical requirement that a Hamiltonian H   be Hermitian is sufficient to guarantee the reality of its eigenvalues, but it is not necessary. By establishing three theorems, this Letter gives conditions that are both necessary and sufficient. First, it is shown that if the secular equation is real, the Hamiltonian is necessarily PTPT symmetric. Second, the set of all operators CC that nontrivially obey the two equations [C,H]=0[C,H]=0 and C2=1C2=1 is introduced. For a diagonalizable PTPT-symmetric Hamiltonian it is shown that the set is nonempty and that the energy eigenvalues of H   are all real only if every such CC commutes with PTPT. There is at least one complex-conjugate pair of eigenvalues whenever at least one such CC does not commute with PTPT. Third, the eigenvalues of any nondiagonalizable Jordan-block matrix that possesses just one eigenvector are all real if the block is PTPT symmetric, and are all complex if it is not. These theorems hold for matrix Hamiltonians of any finite dimensionality.