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Spectral asymptotics for first order systems. (English) Zbl 1423.35281

Summary: This is a review paper outlining recent progress in the spectral analysis of first order systems. We work on a closed manifold and study an elliptic self-adjoint first order system of linear partial differential equations. The aim is to examine the spectrum and derive asymptotic formulae for the two counting functions. Here the two counting functions are those for the positive and the negative eigenvalues. One has to deal with positive and negative eigenvalues separately because the spectrum is, generically, asymmetric.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J46 First-order elliptic systems
35Q41 Time-dependent Schrödinger equations and Dirac equations
35R01 PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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