The spectral function of a first order elliptic system.
Journal of Spectral Theory
7 - 360.
We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the following objects: the propagator (time-dependent operator which solves the Cauchy problem for the dynamic equation), the spectral function (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 lambda) and the counting function (number of eigenvalues between zero and a positive lambda). We derive explicit two-term asymptotic formulae for all three. For the propagator "asymptotic" is understood as asymptotic in terms of smoothness, whereas for the spectral and counting functions "asymptotic" is understood as asymptotic with respect to lambda tending to plus infinity.
|Title:||The spectral function of a first order elliptic system|
|Additional information:||This preprint is "part a" (proper subset) of preprint arXiv:1204.6567. Preprint arXiv:1204.6567 is being split into two parts on the recommendation of the referee of Journal of Spectral Theory. "Part b" will appear as a separate preprint under the title "Spectral theoretic characterization of the massless Dirac operator"|
|Keywords:||math.SP, math.SP, 35P20 (Primary), 35J46, 35R01 (Secondary)|
|UCL classification:||UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics|
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