Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Beyond PT-symmetry: towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

Luís Felipe Alves da Silva

Published Mar 25, 2022• 1 citation
Publisher

Abstract

A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.

DOI
10.11606/d.76.2022.tde-19072022-134347

Related papers

Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

Luís Felipe Alves da Silva, Rodrigo A. Dourado, M. H. Y. Moussa

In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) I ( t ) for a TD non-Hermitian Hamiltonian H(t) H ( t ) . Assuming H(t) H ( t ) to be simultaneously \rho(t) ρ ( t ) -pseudo-Hermitian and \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) I ( t ) = Ξ − 1 ( t ) ρ ( t ) , which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivation of the symmetries associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI Hamiltonians and PT -symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 I 2 k = 1 with k k odd [J. Phys. A 35, L467 (2002)]. The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians. From this relation the TD metric is automatically derived from the TD symmetry of the problem. Then, when placed in perspective with the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) I ( t ) = Ξ − 1 ( t ) ρ ( t ) , the symmetry-metric relation finally allow us to derive the \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond

SciPost Physics Core•Mar 14, 2022•3 citations

On the η pseudo PT symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

Mustapha Maamache

In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{†} via the relation, H^{†}=PTHPT . We propose a derivation of pseudo PT symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric η(t)=PTη(t) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1,1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new metric and discuss a concrete physical applications of our results.

arXiv:2308.13834•Aug 26, 2023•0 citations

On the η pseudo P T symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

Mustapha Maamache

Abstract In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail

Physica Scripta•Dec 19, 2023•0 citations

Pseudo-Hermiticity and generalized PT- and CPT-symmetries

Ali Mostafazadeh

We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (𝒫), time-reversal (𝒯), and charge-conjugation (𝒞) operators and establish the 𝒫𝒯- and 𝒞𝒫𝒯-invariance of H.

arXiv:math-ph/0209018•Journal of Mathematical Physics•Feb 28, 2003•230 citations

On the pseudo-Hermitian invariant method for the time-dependent Non-Hermitian Hamiltonians

Boubakeur Khantoul, Abdelhafid Bounames, Mustapha Maamache

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.

•Oct 28, 2016•3 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
A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.
In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) I ( t ) for a TD non-Hermitian Hamiltonian H(t) H ( t ) . Assuming H(t) H ( t ) to be simultaneously \rho(t) ρ ( t ) -pseudo-Hermitian and \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) I ( t ) = Ξ − 1 ( t ) ρ ( t ) , which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivation of the symmetries associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI Hamiltonians and PT -symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 I 2 k = 1 with k k odd [J. Phys. A 35, L467 (2002)]. The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians. From this relation the TD metric is automatically derived from the TD symmetry of the problem. Then, when placed in perspective with the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t) I ( t ) = Ξ − 1 ( t ) ρ ( t ) , the symmetry-metric relation finally allow us to derive the \Xi(t) Ξ ( t ) -anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond
In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{†} via the relation, H^{†}=PTHPT . We propose a derivation of pseudo PT symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric η(t)=PTη(t) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1,1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new metric and discuss a concrete physical applications of our results.
Abstract In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail
We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (𝒫), time-reversal (𝒯), and charge-conjugation (𝒞) operators and establish the 𝒫𝒯- and 𝒞𝒫𝒯-invariance of H.
We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.
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.