K homology and regular singular Dirac–Schrödinger operators on even-dimensional manifolds (original) (raw)

Index Theory for Perturbed Dirac Operators on Manifolds with Conical Singularities

Proceedings of The American Mathematical Society, 1995

On an odd-dimensional manifold with isolated conical singularities, we perturb a Dirac operator by a vector bundle endomorphism whose pointwise norm grows in inverse proportion to the distance from the singular set. We give two proofs of an index formula for the resulting Fredholm operator. We mention an application to the index theory of transversally elliptic operators.

$L^2$ -index for certain Dirac-Schr�dinger operators

Duke Mathematical Journal, 1992

Dirac-type operator D on a certain class of (noncompact) complete Riemannian manifolds. Although in principle computable, especially in the Fredholm case, this formula contains terms reflecting the contribution of the small eigenvalues, which are difficult to evaluate. We show in this paper that the addition of a skew-adjoint potential V, satisfying reasonable assumptions at infinity, has the effect of eventually overcoming the influence of the small eigenvalues of D. Thus, the L2-index of the "Dirac-Schr/Sdinger operator" D + 2 V, for 2 sufficiently large, is given by an "adiabatic limit" of r/-invariants and is therefore local at infinity. (See Theorem 3.2 below.) This generalizes and at the same time explains index formulae of Callias type.

An Index Formula for Perturbed Dirac Operators on Lie Manifolds

The Journal of Geometric Analysis, 2013

We give an index formula for a class of Dirac operators coupled with unbounded potentials. More precisely, we study operators of the form P := / D + V , where / D is a Dirac operators and V is an unbounded potential at infinity on a possibly noncompact manifold M 0. We assume that M 0 is a Lie manifold with compactification denoted M. Examples of Lie manifolds are provided by asymptotically Euclidean or asymptotically hyperbolic spaces. The potential V is required to be such that V is invertible outside a compact set K and V −1 extends to a smooth function on M K that vanishes on all faces of M in a controlled way. Using tools from analysis on non-compact Riemannian manifolds, we show that the computation of the index of P reduces to the computation of the index of an elliptic pseudodifferential operator of order zero on M 0 that is a multiplication operator at infinity. The index formula for P can then be obtained from the results of [17]. The proof also yields similar index formulas for Dirac operators coupled with bounded potentials that are invertible at infinity on asymptotically commutative Lie manifolds, a class of manifolds that includes the scattering and double-edge calculi.

An index theorem for non-standard Dirac operators

Classical and Quantum Gravity, 1999

On manifolds with non-trivial Killing tensors admitting a square root of the Killing-Yano type one can construct non-standard Dirac operators which differ from, but commute with, the standard Dirac operator. We relate the index problem for the non-standard Dirac operator to that of the standard Dirac operator. This necessitates a study of manifolds with torsion and boundary and we summarize recent results obtained for such manifolds.

Comparison of perturbed Dirac operators

Proceedings of the American Mathematical Society, 1996

This paper extends the index theory of perturbed Dirac operators to a collection of noncompact even-dimensional manifolds that includes both complete and incomplete examples. The index formulas are topological in nature. They can involve a compactly supported standard index form as well as a form associated with a Toeplitz pairing on a hypersurface.

A new method for eigenvalue estimates for Dirac operators on certain manifolds with Sk-symmetry

Differential Geometry and its Applications, 2003

We prove estimates for the first nonnegative eigenvalue of the Dirac operator on certain manifolds with SO(k + 1)-symmetry in terms of geometric properties of the manifold. For the proof we employ an abstract technique which is new in this context and may apply to other cases of manifolds as well.  2002 Published by Elsevier Science B.V. MSC: 58J50; 35P15

The spectrum of twisted Dirac operators on compact flat manifolds

2003

Let M be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of M , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group Z k 2 , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the η-series, in terms of values of Hurwitz zeta functions, and the η-invariant. We give the dimension of the space of harmonic spinors and characterize all Z k 2-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of Z k 2manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat n-manifolds, pairwise nonhomeomorphic to each other of the order of a n .

The Morse and Maslov indices for Schrödinger operators

Journal d'Analyse Mathématique

We study the spectrum of Schrödinger operators with matrix valued potentials utilizing tools from infinite dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with quasi-periodic, Dirichlet and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for the multidimensional Schrödinger operators with periodic potentials. For quasi convex domains in R n we recast the results connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.