Superintegrable and scale invariant quantum systems with position dependent mass

  • A. G. Nikitin Institute of Mathematics of National Academy of Sciences of Ukraine, Kyiv
Keywords: Schrödinger equation


UDC 517.9

Scale invariant Schrödinger equations with position dependent mass admitting second order integrals of motion are classified. 


V. I. Fushchich, 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, no. 4, 519 – 531 (2018),

A. G. Nikitin, Symmetries of Schrodinger equation with scalar and vector potentials, J. Phys. A, 53, no. 45, 455202 (2020),

A. G. Nikitin, Symmetries of the Schrodinger – Pauli equation for neutral particles , J. Math. Phys., 62, no. 8, 083509 (2021),

A. G. Nikitin, Symmetries of the Schrodinger-Pauli equations for charged particles and quasirelativistic Schroodinger equations, J. Phys. A, 55, no. 11, 115202 (2022),

U. Niederer, The maximal kinematical invariance group of the free Schrodinger equations , Helv. Phys. Acta, 45, no. 5, 802 – 810 (1972).

R. L. Anderson, S. Kumei, C. E. Wulfman, Invariants of the equations of wave mechanics. I, Rev. Mex. Fis., 21, no. 1, 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. Uhlíř, I.Friš, Symmetry groups in classical and quantum mechanics, Sov. J. Nucl. Phys., 4, 444 – 450 (1967).

A. Makarov, J. Smorodinsky, Kh. Valiev, P. Winternitz, A systematic search for non-relativistic systems with dynamical symmetries, Nuovo Cim. A, 52, 1061 – 1084 (1967)

Ian Marquette, Pavel Winternitz, Higher order quantum superintegrability: a new Painleve conjecture. Integrability, Supersymmetry and Coherent States, Springer, Cham (2019), pp. 103 – 131.

A. G. Nikitin, Higher-order symmetry operators for Schrodinger equation, In CRM Proceedings and Lecture Notes (AMS), 37, 137 – 144 (2004).

Oldwig von Roos, Position-dependent effective masses in semiconductor theory, Phys. Rev. B, 27, 7547 (1983).

A. de Saavedra, F. Boronat, A. Polls, A. Fabrocini, Effective mass of one He 4 atom in liquid He 3, Phys. Rev. B, 50, 4248 (1994).

P. Harrison, Quantum Wells, Wires and Dots, Wiley, New York (2000).

R. Heinonen, E. G. Kalnins, W. Miller Jr, E. Subag, Structure relations and Darboux contractions for 2D 2nd order superintegrable systems, SIGMA, 11, 043, 33 pp. (2015),

B. K. Berntson, E. G. Kalnins, 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, D. Riglioni, Superintegrable oscillator and Kepler systems on spaces of nonconstant curvature via the Stackel transform , SIGMA, 7, 048 (2011),

O. Ragnisco, D. Riglioni, A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant 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, no. 33, 335201 (2015),

A. G. Nikitin, T. M. Zasadko, Superintegrable systems with position dependent mass, J. Math. Phys., 56, no. 4, 042101 (2015),

A. G. Nikitin, Kinematical invariance groups of the 3d Schrödinger equations with position dependent masses, J. Math. Phys., 58, № 8, 083508, 16 pp. (2017),

A. G. Nikitin, Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix, Ukr. Math. J., 59, no. 3, 439 – 458 (2007),

A. G. Nikitin, V. I. Fushchich, Equations of motion for particles of arbitrary spin invariant under the Galileo group, Theor. and Math. Phys., 44, 584 – 592 (1980)

O. O. Vaneeva, R. O. Popovych, C. Sophocleous, Equivalence transformations in the study of integrability, Physica Scripta, 89, 038003 (2014),

A. G. Nikitin, Generalized Killing tensors of arbitrary valence and order, 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. Ukraine, 16, № 1, 1 – 18 (2019).

How to Cite
Nikitin, A. G. “Superintegrable and Scale Invariant Quantum Systems With Position Dependent Mass”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 3, Apr. 2022, pp. 360-72, doi:10.37863/umzh.v74i3.7162.
Research articles