In superspace a realization of

$\mathfrak {sl}_2$
sl2 is generated by the super Laplace operator and the generalized norm squared. In this paper, an inner product on superspace for which this representation is skew-symmetric is considered. This inner product was already defined for spaces of weighted polynomials (see [K. Coulembier, H. De Bie, and F. Sommen, Orthogonality of Hermite polynomials in superspace and Mehler type formulae, Proc. London Math. Soc. (accepted) arXiv:1002.1118]). In this article, it is proven that this inner product can be extended to the super Schwartz space, but not to the space of square integrable functions. Subsequently, the correct Hilbert space corresponding to this inner product is defined and studied. A complete basis of eigenfunctions for general orthosymplectically invariant quantum problems is constructed for this Hilbert space. Then the integrability of the
$\mathfrak {sl}_2$
sl2-representation is proven. Finally, the Heisenberg uncertainty principle for the super Fourier transform is constructed.

Topics
Lie algebras, Functional equations, General topology, Operator theory, Generalized functions, Chemical elements, Hilbert space, Quantum mechanics, Schrodinger equations, Uncertainty principle
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© 2011 American Institute of Physics.
2011
American Institute of Physics
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