By using the algebraic approach, the Lie symmetries of Schrödinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, and the admissible equivalence relations are clearly indicated. In particular, the Boyer results concerning kinematical invariance groups for arbitrary potentials [C. P. Boyer, Helv. Phys. Acta 47, 450–605 (1974)] are clarified and corrected.
REFERENCES
1.
U.
Niederer
, “The maximal kinematical invariance group of the free Schrödinger equations
,” Helv. Phys. Acta
45
, 802
–810
(1972
).2.
R. L.
Anderson
, S.
Kumei
, and C. E.
Wulfman
, “Invariants of the equations of wave mechanics. I.
,” Rev. Mex. Fis.
21
, 1
–33
(1972
).3.
C. P.
Boyer
, “The maximal kinematical invariance group for an arbitrary potential
,” Helv. Phys. Acta
47
, 450
–605
(1974
).4.
P.
Winternitz
, J.
Smorodinsky
, M.
Uhlir
, and I.
Fris
, “Symmetry groups in classical and quantum mechanics
,” Sov. J. Nucl. Phys.
4
, 444
–450
(1967
).5.
A. A.
Makarov
, J. A.
Smorodinsky
, K.
Valiev
, and P.
Winternitz
, “A systematic search for nonrelativistic systems with dynamical symmetries
,” Nuovo Cimento A
52
, 1061
–1084
(1967
).6.
7.
N. W.
Evans
, “Super-integrability of the Winternitz system
,” Phys. Lett. A
147
, 483
–486
(1990
).8.
N. W.
Evans
, “Superintegrability in classical mechanics
,” Phys. Rev. A
41
, 5666
–5676
(1990
).9.
W.
Miller
, Jr., S.
Post
, and P.
Winternitz
, “Classical and quantum superintegrability with applications
,” J. Phys. A: Math. Theor.
46
, 423001
(2013
).10.
A. G.
Nikitin
, “Supersymmetries in Schrödinger–Pauli equations and in Schrödinger Equations with position dependent mass
,” in Supersymmetry and Coherent States
(Springer
, Cham
, 2019
), pp. 133
–162
.11.
I.
Marquette
and P.
Winternitz
, “Higher order quantum superintegrability: A new ‘Painlevé conjecture’
,” in Integrability, Supersymmetry and Coherent States
(Springer
, Cham
, 2019
), pp. 103
–131
.12.
A. G.
Nikitin
, “Higher-order symmetry operators for Schrödinger equation
,” in CRM Proceedings and Lecture Notes
(AMS
, 2004
), Vol. 37, pp. 137
–144
.13.
C.
Quesne
and V. M.
Tkachuk
, “Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
,” J. Phys. A: Math. Gen.
37
, 4267
(2004
).14.
C.
Quesne
, “Quadratic algebra approach to an exactly solvable position-dependent mass Schrödinger equation in two dimensions
,” SIGMA
3
, 067
(2007
).15.
R.
Koç
and M.
Koca
, “A systematic study on the exact solution of the position dependent mass Schrödinger equation
,” J. Phys. A: Math. Gen.
36
, 8105
(2003
).16.
S. C. Y.
Cruz
and O.
Rosas-Ortiz
, “Dynamical equations, invariants and spectrum generating algebras of mechanical systems with position-dependent mass
,” SIGMA
9
, 004
(2013
).17.
A. P.
Fordy
and Q.
Huang
, “Generalised Darboux–Koenigs metrics and 3-dimensional superintegrable systems
,” SIGMA
15
, 037
(2019
).18.
A. P.
Fordy
and Q.
Huang
, “Superintegrable systems on 3 dimensional flat space
,” J. Geom. Phys.
153
, 103687
(2020
).19.
A. G.
Nikitin
and T. M.
Zasadko
, “Superintegrable systems with position dependent mass
,” J. Math. Phys.
56
, 042101
(2015
).20.
A. G.
Nikitin
and T. M.
Zasadko
, “Group classification of Schrodinger equations with position dependent mass
,” J. Phys. A: Math. Theor.
49
, 365204
(2016
).21.
A. G.
Nikitin
, “Kinematical invariance groups of the 3d Schrödinger equations with position dependent masses
,” J. Math. Phys.
58
, 083508
(2017
).22.
A. G.
Nikitin
, “Superintegrable and shape invariant systems with position dependent mass
,” J. Phys. A: Math. Theor.
48
, 335201
(2015
).23.
A. G.
Nikitin
, “Algebras of discrete symmetries and supersymmetries for the Schroedinger-Pauli equation
,” Int. J. Mod. Phys. A
14
, 885
–897
(1999
).24.
J.
Niederle
and A. G.
Nikitin
, “Extended supersymmetries for the Schroedinger–Pauli equation
,” J. Math. Phys.
40
, 1280
–1293
(1999
).25.
A. G.
Nikitin
and Y.
Karadzhov
, “Enhanced classification of matrix superpotentials
,” J. Phys. A: Math. Gen.
44
, 445202
(2011
).26.
A. G.
Nikitin
, “Matrix superpotentials and superintegrable systems for arbitrary spin
,” J. Phys. A: Math. Theor.
45
, 225205
(2012
).27.
A. G.
Nikitin
, “Superintegrability and supersymmetry of Schrödinger-Pauli equations for neutral particles
,” J. Math. Phys.
53
, 122103
(2012
).28.
A. G.
Nikitin
, “Superintegrable systems with spin invariant with respect to the rotation group
,” J. Phys. A: Math. Theor.
46
, 256204
(2013
).29.
A. G.
Nikitin
, “New exactly solvable systems with Fock symmetry
,” J. Phys. A: Math. Theor.
45
, 485204
(2012
).30.
A. G.
Nikitin
, “Laplace-Runge-Lenz vector for arbitrary spin
,” J. Math. Phys.
54
, 123506
(2013
).31.
A. G.
Nikitin
, “Superintegrable systems with arbitrary spin
,” Ukr. J. Phys.
58
, 1046
–1054
(2013
).32.
A. G.
Nikitin
, “The maximal ‘kinematical’ invariance group for an arbitrary potential revised
,” J. Math. Phys., Anal., Geom.
14
, 519
–531
(2018
).33.
J.
Beckers
, N.
Debergh
, and A. G.
Nikitin
, “On parasupersymmetries and relativistic description for spin one particles: II. The interacting context with (electro)magnetic fields
,” Fortschr. Phys.
43
, 81
–95
(1995
).34.
E.
Ferraro
, N.
Messina
, and A. G.
Nikitin
, “Exactly solvable relativistic model with the anomalous interaction
,” Phys. Rev. A
81
, 042108
(2010
).35.
P.
Olver
, Application of Lie Groups to Differential Equations
(Springer-Verlag
, New York
, 2000
), electronic version: PJ Olver-2000-books.google.com.36.
A. G.
Nikitin
, “Symmetries of Schrödinger equation with scalar and vector potentials
,” J. Phys. A: Math. Theor.
53
, 455202
(2020
).37.
U.
Niederer
, “The maximal kinematical invariance group of the harmonic oscillator
,” Helv. Phys. Acta
47
, 191
–200
(1973
).38.
U.
Niederer
, “The group theoretical equivalence of the free particle, the harmonic oscillator and the free fall
,” in Proceedings of the 2nd International Colloquium on Group Theoretical Methods in Physics
(University of Nijmegen
, The Netherlands
, 1973
).39.
W. I.
Fushchich
and A. G.
Nikitin
, Symmetries of Equations of Quantum Mechanics
(Allerton Press, Inc.
, NY
, 1994
).40.
W. I.
Fushchich
, L. F.
Barannyk
, and A. F.
Barannyk
, “Subgroup analysis of Galilei and Poincare groups and reduction of nonlinear equations
,” (Naukova Dumka
, Kiev
, 1991
) (in Russian), Extended English version of this monograps which includes subalgebras of algebra (3) is unpublished.41.
L.
Ŝnobl
and P.
Winternitz
, Classification and Identification of Lie Algebras
, CRM Monograph Series Vol. 33 (AMS and Centre de Recherches Mathématiques
, 2010
).42.
G. M.
Murakzianov
, “Classification of real structures of Lie algebras of fifth order
,” Izv. Vyssh. Uchebn. Zaved. Mat.
3
, 99
–106
(1963
).43.
P.
Basarab-Horwath
, V.
Lahno
, and R.
Zhdanov
, “The structure of the Lie algebras and the classification problem of partial differential equations
,” Acta Appl. Math.
69
, 43
–94
(2001
).44.
V.
Boyko
, J.
Patera
, and R.
Popovych
, “Computation of invariants of Lie algebras by means of moving frames
,” J. Phys. A: Math. Gen.
39
, 5749
–5762
(2006
).45.
C. M.
Bender
, PT Symmetry in Quantum and Classical Physics
(World Scientific Publishing
, 2018
).46.
A. G.
Nikitin
and R. O.
Popovych
, “Group classification of nonlinear Schrödinger equations
,” Ukr. Math. J.
53
, 1255
–1265
(2001
).47.
A. G.
Nikitin
, “Group classification of systems of non-linear reaction–diffusion equations with general diffusion matrix. I. Generalized Ginsburg–Landau equations
,” J. Math. Anal. Appl.
324
, 615
–628
(2006
).48.
W. I.
Fushchich
and A. G.
Nikitin
, “Higher symmetries and exact solutions of linear and nonlinear Schrödinger equation
,” J. Math. Phys.
38
, 5944
–5959
(1997
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
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