Ordinarily, one imposes the condition of self-adjointness on the
Hamiltonian, representing the mathematical operation of complex
conjugation and matrix transposition. This conventional Hermiticity
condition is sufficient to ensure that the Hamiltonian |

Together with Carl
Bender and others, I have studied such generalization of
Hermiticity using a complex deformation N is a real
parameter. The system exhibits two phases: When N>=2, the
energy spectrum of H is real and positive as a consequence of -symmetry. However, when 1< PTN<2, the spectrum
contains an infinite number of complex eigenvalues and a finite number
of real, positive eigenvalues because -symmetry is
spontaneously broken. The phase transition that occurs at PTN=2 manifests itself in both the quantum-mechanical system and the
underlying -symmetric
classical system. This idea has captured the imagination of many researchers, with regular conferences dedicated to non-hermitian Hamiltonians. PT |

Currently, we are studying applications of -symmetric quasi-exactly solvable model (QES) for
continuous QES-parameter PTJ, see Journal of Physics A 31, L273-L277 (1998).. For integer J, the
lower J eigenstates can be determined exactly
(red). They cross the rest of the spectrum. ] |

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