Positive scalar curvature,higher rho invariants and localization algebras

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick [27, Theorem 1.17].     

Comparison of perturbed Dirac operators

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.

     

On the Functoriality of Lott''s Secondary Analytic Index

For a smooth manifold M, J. Lott has defined a secondary K-group of local systems of R-modules, where R is some ring with a fixed representation . If is a smooth fibre bundle with closed fibres, then he further defines a push-forward . In the present note, using results of X. Ma, we verify that this push-forward is functorial under iterated fibre bundles.     

Equivalence of higher torsion invariants

We show that the smooth torsion of bundles of manifolds constructed by Dwyer, Weiss, and Williams satisfies the axioms for higher torsion developed by Igusa. As a consequence we obtain that the smooth Dwyer–Weiss–Williams torsion is proportional to the higher torsion of Igusa and Klein.     

Rigidity of higher elliptic genera

We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.Both authors are supported in part by NFS     

Heisenberg operators of a Dirac particle interacting with the quantum radiation field

We consider a quantum system of a Dirac particle interacting with the quantum radiation field, where the Dirac particle is in a 4×4-Hermitian matrix-valued potential V. Under the assumption that the total Hamiltonian HV is essentially self-adjoint (we denote its closure by ), we investigate properties of the Heisenberg operator (j=1,2,3) of the j-th position operator of the Dirac particle at time t∈R and its strong derivative dxj(t)/dt (the j-th velocity operator), where xj is the multiplication operator by the j-th coordinate variable xj (the j-th position operator at time t=0). We prove that D(xj), the domain of the position operator xj, is invariant under the action of the unitary operator for all t∈R and establish a mathematically rigorous formula for xj(t). Moreover, we derive asymptotic expansions of Heisenberg operators in the coupling constant q∈R (the electric charge of the Dirac particle).     

Discrete spectrum of electromagnetic Dirac operators

We consider the Dirac operators with electromagnetic fields on 2-dimensional Euclidean space. We offer the sufficient conditions for electromagnetic fields that the associated Dirac operator has only discrete spectrum.

     

Index formulas for geometric Dirac operators in Riemannian foliations

With regards to certain Riemannian foliations we consider Kasparov pairings of leafwise and transverse Dirac operators. Relative to a pairing with a transversal class we commence by establishing an index formula for foliations with leaves of nonpositive sectional curvature. The underlying ideas are then developed in a more general setting leading to pairings of images under the Baum-Connes map in geometricK-theory with transversal classes. Several ideas implicit in the work of Connes and Hilsum-Skandalis are formulated in the context of Riemannian foliations. From these we establish the notion of a dual pairing inK-homology and a theorem of the Grothendieck-Riemann-Roch type.R. G. D. was supported by The National Science Foundation under Grant No. DMS-9304283.J. F. G. and F. W. K. were supported in part by The National Science Foundation under Grant No. DMS-9208182.F. W. K. was also supported in part by an Arnold O. Beckman Research Award from the Research Board of the University of Illinois.     

Spectral asymptotics and regularized traces for Dirac operators on a star-shaped graph

In this article, we consider the spectral problems for Dirac operators on a star-shaped graph. First, we derive the asymptotic expressions of the eigenvalues and then explicitly establish the regularized trace formulae for the operators in terms of the coefficients of the operators.     

Gaussian estimates for elliptic operators with unbounded drift

We consider a strictly elliptic operator

     

A hierarchy of generalized invariants for linear partial differential operators

We study invariants of linear partial differential operators in two variables under gauge transformations. Using the Beals-Kartashova factorization, we construct a hierarchy of generalized invariants for operators of an arbitrary order. We study the properties of these invariants and give some examples. We also show that the classic Laplace invariants correspond to some particular cases of generalized invariants. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 470–478, June, 2006.     

On the structure of spectra of periodic elliptic operators

The paper discusses the problem of absolute continuity of spectra of periodic elliptic operators. A new result on absolute continuity for a matrix operator of Schrödinger type is obtained. It is shown that all types of operators for which the absolute continuity has previously been established can be reduced to this one. It is also discovered that some natural generalizations stumble upon an obstacle in the form of non-triviality of a certain analytic bundle on the two-dimensional torus.

     

Multiplicities of the eigenvalues of periodic Dirac operators

Let us consider the Dirac operator

     

Generic vanishing for harmonic spinors of twisted Dirac operators

In this paper we address the problem of generic vanishing for (negative) harmonic spinors of Dirac operators coupled with variable metric connections.

     

Fundamental solutions for the super Laplace and Dirac operators and all their natural powers

The fundamental solutions of the super Dirac and Laplace operators and their natural powers are determined within the framework of Clifford analysis.     

An approach to the spectrum structure of Dirac operators by the local-compactness method

The purpose of this paper is to investigate the spectra of the Dirac operator . The local compactness of is shown under some assumption on . This method enables us to prove that if as , then and to give a significant sufficient condition that or has a purely discrete spectrum.