It is a well-known fact that on a bounded spectral interval the Dirac spectrum can described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function Z → R and endow the space of all spectra with an arsinh-uniform metric. We prove that the sp...

Let M be a compact Riemannian manifold with an action by isometries of a compact Lie group G. Suppose that this action could be lifted to an action by isometries on a Clifford bundle E over M. We use the method of the Witten deformation to compute the virtual representation-valued index of a transversally elliptic Dirac operator on E. We express the multiplicities of the associated representati...

We study the asymptotic of the spectrum of the spin Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10].

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the Dirac operators, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues.

We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class of spinor fields generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

On a foliated Riemannian manifold (M, gM ,F) with a transverse spin foliation F , we estimate a lower bound for the square of the eigenvalues of the transversal Dirac operator Dtr.

We explicitly determine the high-energy asymptotics for Weyl– Titchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl–Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac oper...

By studying zero modes of the Dirac operator on the lattice, we explicitly construct the Nahm transform of some topologically non-trivial gauge field configurations on the torus.

The Weierstrass representation for surfaces in R3 [8, 13] was generalized for surfaces in R4 in [12] (see also [4]). This paper uses the quaternion language and the explicit formulas for such a representation were written by Konopelchenko in [9] for constructing surfaces which admit soliton deformation governed by the Davey–Stewartson equations. This generalizes his results from [8] where he in...

Table of Contents Introduction—1 §1 The Dirac operator on the Hyperbolic disk A covering of the frame bundle—4 The Dirac operator—7 Covariance of the Dirac operator—10 §2 Local Expansions Eigenfunctions for infinitesimal rotations—14 Differentiating local expansions with respect to the branch points—21 Estimates at infinity—24 §3 L 2 existence results A model for the simply connected covering D...