An Algebraic-Recursive Approach to Generate Higher-Order Symmetry Operators for Schrödinger and Klein-Gordon equations

• URL: http://arxiv.org/abs/2510.22114v1
• Date: Sat, 25 Oct 2025 01:49:23 GMT
• Title: An Algebraic-Recursive Approach to Generate Higher-Order Symmetry Operators for Schrödinger and Klein-Gordon equations
• Authors: Enrique Casanova, Melvin Arias,
• Abstract summary: The work derives first-order symmetry generators, such as translations, rotations, and boosts.<n>The analysis is then extended to higher-order operators, demonstrating how they can be constructed from the first-order ones.<n>This methodology is applied to the Klein-Gordon equation in Minkowski space-time, yielding relativistic symmetry operators.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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 Schr\"odinger 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.

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