Hilbert space for quantum mechanics on superspace

Abstract

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.

References in corpus (7)

• Spherical harmonics and integration in superspace
• Fourier transform and related integral transforms in superspace
• Integration in superspace using distribution theory
• Orthosymplectic Lie superalgebras in superspace analogues of quantum Kepler problems
• Hermite and Gegenbauer polynomials in superspace using Clifford analysis
• Orthosymplectically invariant functions in superspace
• Schrödinger equation with delta potential in superspace

Cited by in corpus (2)

• Clifford algebras, Fourier transforms and quantum mechanics
• Orthogonality of Hermite polynomials in superspace and Mehler type formulae