Number Operators for Riemannian Manifolds

The Dirac operator d+ δ on the Hodge complex of a Riemannian manifold is regarded as an annihilation operator A. On a weighted space L2μΩ, [A,A ] acts as multiplication by a positive constant on excited states if and only if the logarithm of the measure density of dμ satisfies a pair of equations. The equations are equivalent to the existence of a harmonic distance function on M . Under these conditions N = AA has spectrum containing the nonnegative integers. Nonflat, nonproduct examples are given. The results are summarized as a quantum version of the Cheeger–Gromoll splitting theorem. Much of geometric analysis is the study of the “natural second–order differential operator” on a Riemannian manifold, the Laplace–Beltrami operator. In quantum mechanics there is another “natural” second–order differential operator on R, namely the quantum harmonic oscillator H. Up to a constant scaling and shift, H has whole number spectrum, thus it is a number operator. This paper addresses the question of whether R is unique among Riemannian manifolds in this regard. I ask: on a Riemannian manifold (M, g), can one find a “natural” self–adjoint operator which has spectrum Z? Spectrum containing Z? If we look among Schrödinger operators, then the question is that of finding the potential, of course. (See [MT] for this approach on M = R, showing that there are in fact many such potentials.) I approach the answer more restrictively and I look indirectly for a potential, by phrasing the question in terms of a canonical annihilation operator and a measure analogous to the ground state measure. That is, start with a canonical A acting in some (vector–valued) L space of a specified measure. Define A as the adjoint, and define N = AA. Then ask: under what conditions is it true that [A,A] = 1? I study in this paper a particular choice of annihilation operator: A = 2 (