Spectral theoretic characterization of the massless Dirac operator
نویسندگان
چکیده
We consider an elliptic self-adjoint first-order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive λ. We derive an explicit two-term asymptotic formula for the spectral function as λ → +∞, expressing the second asymptotic coefficient via the trace of the subprincipal symbol and the geometric objects encoded within the principal symbol-metric, torsion of the teleparallel connection and topological charge. We then address the question: is our operator a massless Dirac operator on half-densities? We prove that it is a massless Dirac operator on halfdensities if and only if the following two conditions are satisfied at every point of the manifold: (a) the subprincipal symbol is proportional to the identity matrix; and (b) the second asymptotic coefficient of the spectral function is zero. 1. Main results Consider a first-order differential operator A acting on 2-columns v = (v1 v2) of complexvalued half-densities over a connected compact three-dimensional manifold M without boundary. (See [22, Section 1.1.5] for definition of half-density.) We assume the coefficients of the operator A to be infinitely smooth. We also assume that the operator A is formally self-adjoint (symmetric): ∫
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عنوان ژورنال:
- J. London Math. Society
دوره 89 شماره
صفحات -
تاریخ انتشار 2014