Hilbert space for quantum mechanics on superspace. (English) Zbl 1317.35026
Summary: In superspace a realization of \(\mathfrak{sl}_2\) 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 [The authors, and F. Sommen, Proc. Lond. Math. Soc. (3) 103, No. 5, 786–825 (2011; Zbl 1233.33004)]). 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 \(\mathfrak{sl}_2\)-representation is proven. Finally, the Heisenberg uncertainty principle for the super Fourier transform is constructed.{
©2011 American Institute of Physics}
©2011 American Institute of Physics}
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 46S60 | Functional analysis on superspaces (supermanifolds) or graded spaces |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
Citations:
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