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2012 | 10 | 3 | 927-941
Tytuł artykułu

Global $\widetilde{SL(2,R)}$ representations of the Schrödinger equation with singular potential

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Języki publikacji
EN
Abstrakty
EN
We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V λ(x) = λx −2 as a representation of $\widetilde{SL(2,\mathbb{R})}$ . The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of $\widetilde{SL(2,\mathbb{R})}$ ⋉ H 3, where H 3 is the three-dimensional Heisenberg group.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
3
Strony
927-941
Opis fizyczny
Daty
wydano
2012-06-01
online
2012-03-24
Twórcy
autor
Bibliografia
  • [1] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, DC, 1964
  • [2] Boyer C.P., The maximal ‘kinematical’ invariance group for an arbitrary potential, Helv. Phys. Acta, 1974, 47(5), 589–605
  • [3] Coddington E.A., An Introduction to Ordinary Differential Equations, Prentice-Hall Mathematics Series, Prentice-Hall, Englewood Cliffs, 1961
  • [4] Craddock M.J., Dooley A.H., On the equivalence of Lie symmetries and group representations, J. Differential Equations, 2010, 249(3), 621–653 http://dx.doi.org/10.1016/j.jde.2010.02.003
  • [5] Galajinsky A., Lechtenfeld O., Polovnikov K., Calogero models and nonlocal conformal transformations, Phys. Lett. B, 2006, 643(3–4), 221–227
  • [6] Kashiwara M., Vergne M., On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math., 1978, 44(1), 1–47 http://dx.doi.org/10.1007/BF01389900
  • [7] Niederer U., The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta, 1972, 45(5), 802–810
  • [8] Sepanski M.R., Stanke R.J., Global Lie symmetries of the heat and Schrödinger equation, J. Lie Theory, 2010, 20(3), 543–580
  • [9] Truax D.R., Symmetry of time-dependent Schrödinger equations I. A classification of time-dependent potentials by their maximal kinematical algebras, J. Math. Phys., 1981, 22(9), 1959–1964 http://dx.doi.org/10.1063/1.525142
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0040-8
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