Fang, Yan-Long; (2017) Analysis of first order systems on manifolds without boundary: A spectral theoretic approach. Doctoral thesis , UCL (University College London).

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Abstract

In this thesis we study first order systems of partial differential equations on manifolds without boundary. The thesis is, in part, based upon my publications. The importance of analysing first order systems can be seen directly from the special case, namely, the study of Dirac-type operators. In the first part, I assume my manifolds to be 4-dimensional and consider the simplest non-trivial first order linear differential operators on such manifolds. I give a systematic way of extracting the geometric content encoded within these operators. More importantly, a new concept called covariant subprincipal symbol is defined and further employed in the spectral analysis developed in the next chapter. In the second part, by applying the hyperbolic equation method and using Fourier Tauberian theorems, I establish the relationship between the Weyl coefficients of an elliptic self-adjoint first order differential operator and the residues of the corresponding eta function, which can be easily generalised to the pseudo-differential case. The special case of this relationship is examined explicitly and it is combined with the analysis of the first part. The third part of the thesis involves a detailed analysis of the massless Dirac operator, whereas the massive case is also investigated with the help of the abstract adjugation operation on operators. In contrast to the asymptotics of large eigenvalues, there has not been a systematic and robust way of analysing the asymptotics of small eigenvalues. The last part of the thesis gives a rigorous perturbation analysis of the massless Dirac operator on a topological 3-sphere, which complements the results obtained in the second part. In particular, explicit asymptotic formulae for small eigenvalues are derived for general perturbations of the standard metric. These asymptotic formulae are tested on generalised Berger spheres for which we have explicit expressions for eigenvalues.

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