Global S L ( 2 , R ) ˜ representations of the Schrödinger equation with singular potential

Jose Franco

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 927-941
  • ISSN: 2391-5455

Abstract

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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 S L ( 2 , 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 S L ( 2 , ℝ ) ˜ ⋉ H 3, where H 3 is the three-dimensional Heisenberg group.

How to cite

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Jose Franco. "Global $\widetilde{SL(2,R)}$ representations of the Schrödinger equation with singular potential." Open Mathematics 10.3 (2012): 927-941. <http://eudml.org/doc/269457>.

@article{JoseFranco2012,
abstract = {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.},
author = {Jose Franco},
journal = {Open Mathematics},
keywords = {Schrödinger equation; Time-dependent potentials; Lie theory; Representation theory; Globalizations; time-dependent potentials; representation theory; globalizations},
language = {eng},
number = {3},
pages = {927-941},
title = {Global $\widetilde\{SL(2,R)\}$ representations of the Schrödinger equation with singular potential},
url = {http://eudml.org/doc/269457},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Jose Franco
TI - Global $\widetilde{SL(2,R)}$ representations of the Schrödinger equation with singular potential
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 927
EP - 941
AB - 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.
LA - eng
KW - Schrödinger equation; Time-dependent potentials; Lie theory; Representation theory; Globalizations; time-dependent potentials; representation theory; globalizations
UR - http://eudml.org/doc/269457
ER -

References

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  5. [5] Galajinsky A., Lechtenfeld O., Polovnikov K., Calogero models and nonlocal conformal transformations, Phys. Lett. B, 2006, 643(3–4), 221–227 Zbl1248.81105
  6. [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 Zbl0375.22009
  7. [7] Niederer U., The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta, 1972, 45(5), 802–810 
  8. [8] Sepanski M.R., Stanke R.J., Global Lie symmetries of the heat and Schrödinger equation, J. Lie Theory, 2010, 20(3), 543–580 Zbl05792649
  9. [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 Zbl0475.35038

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