Résumé:
In this thesis we have discussed on one of the recent topics in physics,
in particular in quantum mechanics, which is "the non-Hermitian Hamiltonians",
and the conditions that make the spectra of these Hamiltonians real.
Initially, the Hermeticity was a necessary and su cient condition for the reality
of the Hamiltonian spectrum. Then a new quantum theory called the
"PT -symmetry" started to emerge, this theory was developed in 1998 by
Carl Bender and Stefan Boettcher, where they revealed the existence of a
class of non-Hermitian Hamiltonians with real spectra. These Hamiltonians
are invariant under the transformation of PT -symmetry, where P is the parity
operator and T is the time reversal operator. A few years later, another
alternative approach was developed in 2002 by Mostafazadeh who works on
pseudo-Hermitian Hamiltonians and he showed that every Hamiltonian with
a real spectrum is pseudo-Hermitian. We have found from an application on
two examples of a pseudo-Hermitian Hamiltonians " shifted harmonic oscillator
" and " cubic anharmonic oscillator " that the energy spectrum of these
Hamiltonians are real and positive.
Keywords: Hermiticity, PT -symmetry, Pseudo-Hermiticity, Quasi-Hermiticity.
