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A unified approach to Hermitian and Unitary operators

Chandana Roy

Published Jan 1, 1977• 0 citations
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Basic properties of certain types of operators: A unified treatment

Chandana Roy

Using normal operators as a common footing, the basic properties of Hermitian operators, unitary operators, and anti-Hermitian operators are treated from a unified viewpoint. The interlinks among the eigenvalues of these operators are elucidated with the help of suitable prescriptions for mapping in the plane of complex eigenvalues. Further, we show that some of the characteristics of Hermitian operators hold good under a perspective which is broader than the defining criterion for Hermitian operators.

American Journal of Physics•Apr 1, 1978•0 citations

Non-Hermitian Hamiltonian beyond PT symmetry for time-dependent S U ( 1 , 1 ) and S U ( 2 ) systems — Exact solution and geometric phase in pseudo-invariant theory

Nadjat Amaouche, Maroua Sekhri, Rahma Zerimeche +2 more

In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.

Physics Open•Nov 21, 2022•4 citations
Using normal operators as a common footing, the basic properties of Hermitian operators, unitary operators, and anti-Hermitian operators are treated from a unified viewpoint. The interlinks among the eigenvalues of these operators are elucidated with the help of suitable prescriptions for mapping in the plane of complex eigenvalues. Further, we show that some of the characteristics of Hermitian operators hold good under a perspective which is broader than the defining criterion for Hermitian operators.
In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.