The Maximal "Kinematical" Invariance Group for an Arbitrary Potential Revised
DOI:
https://doi.org/10.15407/mag14.04.519Ключові слова:
рівняння Шредінгера, симетрія Лі, перетворення еквівалентності.Анотація
Переглянуто групову класифікацію одночастинкового рівняння Шредінгера з довільним потенціалом (C.P. Boyer, Helv. Phys. Acta 47 (1974), p. 450). Представлено виправлений перелік нееквівалентних потенціалів та відповідних симетрій разом з точною ідентифікацією алгебр симетрій та допустимих перетворень еквівалентності.Mathematics Subject Classification: 34L15, 34L20, 35R10.
Посилання
R.L. Anderson, S. Kumei, and C.E. Wulfman, Invariants of the equations of wave mechanics. I, Rev. Mexicana Fı́s. 21 (1972), 1–33.
P. Basarab-Horwath, L. Lahno, and R. Zhdanov, The structure of the Lie algebras and the classification problem of partial differential equations, Acta Appl. Math. 69 (2001), 43–94. https://doi.org/10.1023/A:1012667617936
C.P. Boyer, The maximal ‘kinematical’ invariance group for an arbitrary potential, Helv. Phys. Acta 47 (1974), 450–605.
V. Boyko, J. Patera, and R. Popovych, Computation of invariants of Lie algebras by means of moving frames, J. Phys. A 39 (2006), 5749–5762. https://doi.org/10.1088/0305-4470/39/20/009
W.I. Fushchich, L.F. Barannyk, and A.F. Barannyk, Subgroup Analysis of Galilei and Poincare Groups and Reduction of Nonlinear Equations (Russian), Naukova Dumka, Kiev, 1991.
W.I. Fushchich and A.G. Nikitin, Conformal invariance of relativistic equations for arbitrary spin particles, Lett. Math. Phys. 2 (1977/78), 471–475. https://doi.org/10.1007/BF00398499
W.I. Fushchich and A.G. Nikitin, Higher symmetries and exact solutions of linear and nonlinear Schrödinger equation, J. Math. Phys. 38 (1997), 5944–5959. https://doi.org/10.1063/1.532180
C.R. Hagen, Scale and conformal transformations in Galilean-invariant conformal field theory, Phys. Rev. D5 (1972), 377–388. https://doi.org/10.1103/PhysRevD.5.377
J.-M. Levy-Leblond, Galilei group and non-relativistic quantum mechanics, J. Math. Phys. 4 (1963), 776–788. https://doi.org/10.1063/1.1724319
W. Miller, Jr., Symmetry and Separation of Variables, Encyclopedia of Mathematics and its Applications, 4, Addison–Wesley Publishing Co., Reading, Mass.–London– Amsterdam, 1977.
G.M. Murakzianov, Classification of real structures of Lie algebras of fifth order, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (1963), 99–106 (Russian).
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equations, Helv. Phys. Acta 45 (1972), 802–810.
U. Niederer, The maximal kinematical invariance group of the harmonic oscillator, Helv. Phys. Acta 47 (1973), 191–200.
U. Niederer, The Group Theoretical Equivalence of the Free Particle, the Harmonic Oscillator and the Free Fall, Proceedings of the 2nd International Colloquium on Group Theoretical Methods in Physics, University of Nijmegen, The Netherlands, 1973.
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 (2006), 615–628. https://doi.org/10.1016/j.jmaa.2005.12.022
A.G. Nikitin, Kinematical invariance groups of the 3d Schrödinger equations with position dependent masses, J. Math. Phys. 58 (2017), 083508. https://doi.org/10.1063/1.4986171
A.G. Nikitin and R.O. Popovich, Group classification of nonlinear Schrödinger equations, Ukrainian Math. J. 53 (2001), 1255–1265. https://doi.org/10.1023/A:1013347626895
A.G. Nikitin and T.M. Zasadko, Superintegrable systems with position dependent mass, J. Math. Phys. 56 (2015), 042101. https://doi.org/10.1063/1.4908107
A.G. Nikitin and T.M. Zasadko, Group classification of Schrödinger equations with position dependent mass, J. Phys. A: Math. Theor. 49 (2016), 365204. https://doi.org/10.1088/1751-8113/49/36/365204
R.O. Popovych, V.M. Boyko, M.O. Nesterenko, and M.W. Lutfullin, Realizations of real low-dimensional Lie algebras, J. Phys. A 36 (2003), 7337–7360. https://doi.org/10.1088/0305-4470/36/26/309
L. S̆nobl and P. Winternitz. Classification and Identification of Lie Algebras, CRM Monograph Series, 33, American Mathematical Society, Providence, RI, 2014. https://doi.org/10.1090/crmm/033
P. Winternitz, Ya.A. Smorodinskiı̆, M. Uhlir̆, and I. Fris̆, Symmetry groups in classical and quantum mechanics, Yad. Fiz. 4 (1966), 625–635,(Russian); Engl. transl.: Nucl. Phys. 4 (1967), 444–450.
