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

• Autor(a) principal: Silva, Luís Felipe Alves da
• Data de Publicação: 2022
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/76/76134/tde-19072022-134347/
• topic: General antilinear symmetry; Pseudo-hermiticidade DT; PT-simetria; PT-symmetry; Simetria antilinear geral; TD pseudo-Hermiticity

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: I2k = 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. Our results reinforce the prospects of going beyond PT-symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.