Renormalized oscillation theory for Dirac operators

Oscillation theory for one-dimensional Dirac operators with separated boundary conditions is investigated. Our main theorem reads: If and if solve the Dirac equation , (in the weak sense) and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection equals the number of zeros of the Wronskian of and . As an application we establish finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.

     

Relative oscillation theory for Sturm-Liouville operators extended

We extend relative oscillation theory to the case of Sturm-Liouville operators Hu=r⁻¹(−^′(pu^′)+qu) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.     

Deforming the point spectra of one-dimensional Dirac operators

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operators are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the original operator to its deformed version is explicitly determined.

     

Bismut superconnections and the Chern character for Dirac operators on foliated manifolds

In this paper, we show how to define a Bismut superconnection for generalized Dirac operators defined along the leaves of a compact foliated manifoldM. Using the heat operator of the curvature of the superconnection, we define a (nonnormalized) Chern character for the Dirac operator, which lies in the Haefliger cohomology of the foliation. Rescaling the metric onM by 1/a and lettinga 0, we obtain the analog of the classical cohomological formula for the index of a family of Dirac operators. In certain special cases, we can also compute the limit asa and show that it is the Chern character of the index bundle given by the kernel of the Dirac operator. Finally, we discuss the relation of our results with the Chern character in cyclic cohomology.     

Vanishing theorem for transverse Dirac operators on Riemannian foliations

We obtain a vanishing theorem for the half-kernel of a transverse Spin ^c Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation twisted by a sufficiently large power of a line bundle, whose curvature vanishes along the leaves and is transversely non-degenerate at any point of the ambient manifold.      

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.

     

Spectral bounds for Dirac operators on open manifolds

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the author’s estimate on surfaces.      

On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators

In this paper we consider periodic Dirac operators with skew-adjoint potentials in a large class of weighted Sobolev spaces. We characterize the smoothness of such potentials by asymptotic properties of the periodic spectrum of the corresponding Dirac operators.     

The Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ? ? ⁿ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function u in a Sobolev space H ^s (D), s ∈ ?, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ?D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.     

On fractional semidiscrete Dirac operators of Lévy–Leblond type

In this paper, we introduce a wide class of space-fractional and time-fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space-time lattice $h{\mathbb{Z}}^n\times [0,\infty )$ ($h>0$), resembling to fractional semidiscrete counterparts of the so-called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup ${\left\{\exp (-t{e}^{i\theta }{(-{\mathrm{\Delta }}_h)}^{\alpha })\right\}}_{t\ge 0}$, carrying the parameter constraints and $|\theta |\le \frac{\alpha \pi }{2}$. The results obtained involve the study of Cauchy problems on $h{\mathbb{Z}}^n\times [0,\infty )$.     

Resonances for Dirac operators on the half-line

We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain.     

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.     

Resonances for 1D massless Dirac operators

We consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain, (3) the trace formula in terms of resonances.     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

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.