In this article, we survey the gluing formulae of the spectral invariants the ζ-regularized determinant of a Laplace type operator and the eta invariant of a Dirac type operator. After these spectral invariants had been originally introduced by Ray-Singer [30] and Atiyah-Patodi-Singer [1] respectively, these invariants have been studied by many people in many different parts of mathematics and ...

We present a matrix technique to obtain the spectrum and the analytical index of some elliptic operators defined on compact Riemannian manifolds. The method uses matrix representations of the derivative which yield exact values for the derivative of a trigono-metric polynomial. These matrices can be used to find the exact spectrum of an elliptic operator in particular cases and in general, to g...

Abstract. We first give an elementary proof of a result relating the eigenvalues of the Dirac operator to Branson’s Q-curvature on 4-dimensional spin compact manifolds. In the case of n-dimensional closed compact (spin) manifolds we then use the conformal covariance of the Dirac, Yamabe and Branson-Paneitz operators to compare appropriate powers of their first eigenvalues. Equality cases are al...

§

In this paper we define a one-dimensional discrete Dirac operator on Z. We study the eigenvalues of the Dirac operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity.

We establish a general splitting formula for index bundles of families of Dirac type operators. Among the applications, our result provides a positive answer to a question of Bismut and Cheeger [BC2].

I will show that the first five axioms I had given in '96 on spectral triples suffice in the com-mutative case to characterize smooth compact manifolds. I will also define a new invariant in Riemannian geometry, which when combined with the spectrum of the Dirac operator is a complete invariant of the geometry. It is an analogue of the CKM mixing matrix of the Standard model.

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension ≥ 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension ≥ 3 on a ...

We show that the Dirac operator on a spin manifold does not admit L eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.

The dynamical group associated with the Dirac equation with a radially symmetric potential in four space-time dimensions is represented in terms of integrals with respect to operator valued set functions associated with the free Dirac operator. In coordinates in which c = ~ = 1, the class of potentials treated includes Coulomb potentials ?a=r with jaj < 1.