  In superspace a realization of 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, 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 sl2-representation is proven. Finally the Heisenberg uncertainty
principle for the super Fourier transform is constructed.
