Bibliographic Information: Nikitin, A. G. (2024). Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 2. Systems with dilatation and shift symmetries. arXiv preprint arXiv:2407.20112v2.
Research Objective: This research aims to classify three-dimensional quantum mechanical systems with position-dependent masses (PDM) that possess at least one second-order integral of motion and exhibit invariance under either dilatation or shift transformations.
Methodology: The authors employ an optimized algorithm to solve the determining equations for second-order integrals of motion. They leverage the invariance of the considered PDM systems under dilatation and shift transformations to simplify the determining equations and achieve a classification.
Key Findings: The study identifies nine inequivalent PDM systems with dilatation symmetry, categorized as four integrable, three superintegrable, and two maximally superintegrable systems. Additionally, it reveals eighteen inequivalent systems invariant under shift transformations, comprising seven integrable, seven superintegrable, and four maximally superintegrable systems.
Main Conclusions: The classification demonstrates that a significant number of PDM systems exist that possess second-order integrals of motion and are invariant under dilatation or shift transformations. The majority of these identified systems are novel and have not been previously discovered using traditional approaches for classifying superintegrable systems.
Significance: This research significantly contributes to the understanding of integrable and superintegrable systems in quantum mechanics, particularly in the context of position-dependent masses. The identified systems and the classification methodology hold potential for applications in various areas of theoretical physics where PDM systems are relevant.
Limitations and Future Research: The study focuses specifically on systems with dilatation and shift symmetries. Further research could explore other one-parameter Lie symmetry groups admitted by PDM Schrödinger equations, such as rotations around a fixed axis and specific combinations of transformations. Additionally, investigating the physical implications and potential applications of the classified systems in different physical models would be a valuable avenue for future work.
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by A. G. Nikiti... um arxiv.org 10-11-2024
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