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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2510.22114 |
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| _version_ | 1866915576617631744 |
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| author | Casanova, Enrique Arias, Melvin |
| author_facet | Casanova, Enrique Arias, Melvin |
| contents | This article explores an algebraic-recursive approach to construct differential operators that commute with a central operator $\hat{H}$ in quantum mechanics. Starting from the Schrödinger equation for a free particle, the work derives first-order symmetry generators, such as translations, rotations, and boosts, and examines their algebraic basis encompassing Lie and Jordan algebras. The analysis is then extended to higher-order operators, demonstrating how they can be constructed from the first-order ones through algebraic operations and Lie algebra simplification. This methodology is applied to the Klein-Gordon equation in Minkowski space-time, yielding relativistic symmetry operators. Furthermore, we defined an approximation to fractional symmetry operators of the Schrodinger equation, and a perturbative approach is employed for a case where the commutation is more general, illustrated with a one-dimensional harmonic oscillator and the fourth-order Klein-Gordon equation. The results include a general formula for the number of operators as a function of the order and the dimension of the algebraic basis, providing a reduced-form development of the differential higher-order centralizers' basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Algebraic-Recursive Approach to Generate Higher-Order Symmetry Operators for Schrödinger and Klein-Gordon equations Casanova, Enrique Arias, Melvin Quantum Physics Mathematical Physics This article explores an algebraic-recursive approach to construct differential operators that commute with a central operator $\hat{H}$ in quantum mechanics. Starting from the Schrödinger equation for a free particle, the work derives first-order symmetry generators, such as translations, rotations, and boosts, and examines their algebraic basis encompassing Lie and Jordan algebras. The analysis is then extended to higher-order operators, demonstrating how they can be constructed from the first-order ones through algebraic operations and Lie algebra simplification. This methodology is applied to the Klein-Gordon equation in Minkowski space-time, yielding relativistic symmetry operators. Furthermore, we defined an approximation to fractional symmetry operators of the Schrodinger equation, and a perturbative approach is employed for a case where the commutation is more general, illustrated with a one-dimensional harmonic oscillator and the fourth-order Klein-Gordon equation. The results include a general formula for the number of operators as a function of the order and the dimension of the algebraic basis, providing a reduced-form development of the differential higher-order centralizers' basis. |
| title | An Algebraic-Recursive Approach to Generate Higher-Order Symmetry Operators for Schrödinger and Klein-Gordon equations |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2510.22114 |