Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians
In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for a TD non-Hermitian Hamiltonian H(t). Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symm...
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| Format: | Article |
| Language: | English |
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SciPost
2022-03-01
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| Series: | SciPost Physics Core |
| Online Access: | https://scipost.org/SciPostPhysCore.5.1.012 |
| _version_ | 1818141663850135552 |
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| author | Luís F. Alves da Silva, Rodrigo A. Dourado, Miled H. Y. Moussa |
| author_facet | Luís F. Alves da Silva, Rodrigo A. Dourado, Miled H. Y. Moussa |
| author_sort | Luís F. Alves da Silva, Rodrigo A. Dourado, Miled H. Y. Moussa |
| collection | DOAJ |
| description | In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for a TD non-Hermitian Hamiltonian H(t). 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 derivaton of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI lHamiltonians and PT-symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 with 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)=Ξ⁻¹(t)ρ(t), the symmetry-metric relation finally allow us to derive the Ξ(t)-anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond PT-symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising. |
| first_indexed | 2024-12-11T11:03:28Z |
| format | Article |
| id | doaj.art-2c7f87ee7a89434198b61411629f6a96 |
| institution | Directory Open Access Journal |
| issn | 2666-9366 |
| language | English |
| last_indexed | 2024-12-11T11:03:28Z |
| publishDate | 2022-03-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics Core |
| spelling | doaj.art-2c7f87ee7a89434198b61411629f6a962022-12-22T01:09:47ZengSciPostSciPost Physics Core2666-93662022-03-015101210.21468/SciPostPhysCore.5.1.012Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian HamiltoniansLuís F. Alves da Silva, Rodrigo A. Dourado, Miled H. Y. MoussaIn this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for a TD non-Hermitian Hamiltonian H(t). 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 derivaton of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI lHamiltonians and PT-symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1 with 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)=Ξ⁻¹(t)ρ(t), the symmetry-metric relation finally allow us to derive the Ξ(t)-anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond PT-symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.https://scipost.org/SciPostPhysCore.5.1.012 |
| spellingShingle | Luís F. Alves da Silva, Rodrigo A. Dourado, Miled H. Y. Moussa Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians SciPost Physics Core |
| title | Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians |
| title_full | Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians |
| title_fullStr | Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians |
| title_full_unstemmed | Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians |
| title_short | Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians |
| title_sort | beyond pt symmetry towards a symmetry metric relation for time dependent non hermitian hamiltonians |
| url | https://scipost.org/SciPostPhysCore.5.1.012 |
| work_keys_str_mv | AT luisfalvesdasilvarodrigoadouradomiledhymoussa beyondptsymmetrytowardsasymmetrymetricrelationfortimedependentnonhermitianhamiltonians |