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Real Spectra in Non-Hermitian Hamiltonians HavingPPT T Symmetry The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below Replacing this condition by the weaker condition of PT symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive These PT symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase