Pseudo-Hermiticity versus PT Symmetry: The structure responsible for the reality of the spectrum of a non-Hermitian Hamiltonian
نویسنده
چکیده
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT -symmetric non-Hermitian Hamiltonians studied in the literature are pseudo-Hermitian and argue that the structure responsible for the reality of the spectrum of these Hamiltonian is their pseudo-Hermiticity not PT -symmetry. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRWmodels coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
منابع مشابه
Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT -symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermit...
متن کاملPseudo-Hermiticity versus PT -Symmetry II: A complete characterization of non-Hermitian Hamiltonians with a real spectrum
We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors. Recently, we have explored in [1] the basic mathematical structure underlying the spectral properties of PT -symmetric Hamiltonians [2]. In particular, we have shown that these properties are associated with a class of more general (...
متن کاملPseudo-Hermiticity versus PT Symmetry III: Equivalence of pseudo-Hermiticity and the presence of anti-linear symmetries
We show that a (non-Hermitian) Hamiltonian H admitting a complete biorthonormal set of eigenvectors is pseudo-Hermitian if and only if it has an anti-linear symmetry, i.e., a symmetry generated by an anti-linear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H ...
متن کاملPseudo-reality and pseudo-adjointness of Hamiltonians
We define pseudo-reality and pseudo-adjointness of a Hamiltonian, H, as ρHρ−1 = H∗ and μHμ−1 = H ′, respectively. We prove that the former yields the necessary condition for spectrum to be real whereas the latter helps in fixing a definition for inner-product of the eigenstates. Here we separate out adjointness of an operator from its Hermitian-adjointness. It turns out that a Hamiltonian posse...
متن کاملExact PT-Symmetry Is Equivalent to Hermiticity
We show that a quantum system possessing an exact antilinear symmetry, in particular PT -symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT -symmetry to a Hermitian Hamiltonian. We apply our general results to PT symmetry in finite-dimensions and give the explicit form of the...
متن کامل