We present a classification of the scale-invariant Schrödinger equations with position-dependent mass admitting second-order integrals of motion.
Similar content being viewed by others
References
V. I. Fushchich and A. G. Nikitin, Symmetries of Equations of Quantum Mechanics, Allerton Press, New York (1994).
A. G. Nikitin, “The maximal ‘kinematical’ invariance group for an arbitrary potential revised,” J. Math. Phys., Analysis, Geometry 14, 519–531 (2018).
A. G. Nikitin, “Symmetries of Schrödinger equation with scalar and vector potentials,” J. Phys. A, 53, 455202 (2020).
A. G. Nikitin, “Symmetries of the Schrödinger–Pauli equation for neutral particles,” J. Math. Phys., 62, 083509 (2021).
A. G. Nikitin, “Symmetries of the Schrödinger–Pauli equations for charged particles and quasirelativistic Schrödinger equations,” J. Phys. A, 55, 115202 (2022).
U. Niederer, “The maximal kinematical invariance group of the free Schrödinger equations,” Helv. Phys. Acta, 45, 802–810 (1972).
R. L. Anderson, S. Kumei, and C. E. Wulfman, “Invariants of the equations of wave mechanics. I,” Rev. Mex. Fis., 21, 1–33 (1972).
C. P. Boyer, “The maximal kinematical invariance group for an arbitrary potential,” Helv. Phys. Acta, 47, 450–605 (1974).
P. Winternitz, J. Smorodinsky, M. Uhliř, and I. Friš, “Symmetry groups in classical and quantum mechanics,” Sov. J. Nucl. Phys., 4, 444–450 (1967).
A. Makarov, J. Smorodinsky, Kh. Valiev, and P.Winternitz, “A systematic search for non-relativistic systems with dynamical symmetries,” Nuovo Cim. A, 52, 1061–1084 (1967).
I. Marquette and P.Winternitz, “Higher order quantum superintegrability: a new Painleve conjecture. Integrability,” in: Supersymmetry and Coherent States, Springer, Cham (2019), pp. 103–131.
A. G. Nikitin, “Higher-order symmetry operators for Schrödinger equation,” in: CRM Proceedings and Lecture Notes (AMS), 37 (2004), pp. 137–144.
O. von Roos, “Position-dependent effective masses in semiconductor theory,” Phys. Rev. B, 27, 7547 (1983).
A. de Saavedra, F. Boronat, A. Polls, and A. Fabrocini, “Effective mass of one He 4 atom in liquid He 3,” Phys. Rev. B, 50, 4248 (1994).
P. Harrison, Quantum Wells, Wires,s and Dots, Wiley, New York (2000).
R. Heinonen, E. G. Kalnins, W. Miller Jr., and E. Subag, “Structure relations and Darboux contractions for 2D 2nd order superintegrable systems,” SIGMA, 11, 043 (2015).
B. K. Berntson, E. G. Kalnins, and W. Miller Jr., “Toward classification of 2nd order superintegrable systems in 3-dimensional conformally flat spaces with functionally linearly dependent symmetry operators,” SIGMA: Symmetry, Integrability and Geometry: Methods and Applications, 16, 135 (2020).
A. Ballesteros, A. Enciso, F. J. Herranz, O. Ragnisco, and D. Riglioni, “Superintegrable oscillator and Kepler systems on spaces of nonconstant curvature via the Stäckel transform,” SIGMA, 7, 048 (2011).
O. Ragnisco and D. Riglioni, “A family of exactly solvable radial quantum systems on space of nonconstant curvature with accidental degeneracy in the spectrum,” SIGMA, 6, 097 (2010).
A. G. Nikitin, “Superintegrable and shape invariant systems with position dependent mass,” J. Phys. A: Math. Theor., 48, 335201 (2015).
A. G. Nikitin and T. M. Zasadko, “Superintegrable systems with position dependent mass,” J. Math. Phys., 56, 042101 (2015).
A. G. Nikitin, “Kinematical invariance groups of the 3d Schrödinger equations with position dependent masses,” J. Math. Phys., 58, No. 8, 083508 (2017).
A. G. Nikitin, “Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix,” Ukr. Mat. Zh., 59, No. 3, 395–411 (2007)); English translation: Ukr. Math. J., 59, No. 3, 439–458 (2007).
A. G. Nikitin and V. I. Fushchich, “Equations of motion for particles with arbitrary spin invariant under the Galileo group,” Theor. Math. Phys., 44, 584–592 (1980)
O. O. Vaneeva, R. O. Popovych, and C. Sophocleous, “Equivalence transformations in the study of integrability,” Physica Scripta, 89, 038003 (2014).
A. G. Nikitin, “Generalized Killing tensors of arbitrary rank and order,” Ukr. Mat. Zh., 43, No. 6, 786–795 (1991); English translation: Ukr. Math. J., 43, No. 6, 734–743 (1991).
A. G. Nikitin, “Exact solvability of PDM systems with extended Lie symmetries,” Proc. Inst. Math. Nat. Acad. Sci. Ukr., 16, No. 1, 1–18 (2019).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 3, pp. 360–372, March, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i3.7162.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Nikitin, A.G. Superintegrable and Scale-Invariant Quantum Systems with Position-Dependent Mass. Ukr Math J 74, 405–419 (2022). https://doi.org/10.1007/s11253-022-02072-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02072-8