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1.
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. 相似文献
2.
Robert Stadler 《Journal of Mathematical Analysis and Applications》2010,371(2):638-648
We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators. 相似文献
3.
Gerald Teschl 《Proceedings of the American Mathematical Society》1998,126(6):1685-1695
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.
4.
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. 相似文献
6.
James L. Heitsch 《K-Theory》1995,9(6):507-528
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. 相似文献
7.
Gerald Teschl 《Proceedings of the American Mathematical Society》1998,126(10):2873-2881
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.
8.
Naohiro Suzuki 《Proceedings of the American Mathematical Society》2000,128(3):819-825
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.
9.
10.
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. 相似文献
11.
Yuri A. Kordyukov 《Annals of Global Analysis and Geometry》2008,34(2):195-211
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.
相似文献
12.
Christian Bär 《Annals of Global Analysis and Geometry》2009,36(1):67-79
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.
相似文献
13.
Roberto J. Miatello Ricardo A. Podestá 《Transactions of the American Mathematical Society》2006,358(10):4569-4603
Let 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 , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the -series, in terms of values of Hurwitz zeta functions, and the -invariant. We give the dimension of the space of harmonic spinors and characterize all -manifolds having asymmetric Dirac spectrum.
Furthermore, we exhibit many examples of Dirac isospectral pairs of -manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat -manifolds, pairwise nonhomeomorphic to each other of the order of .
14.
15.
Nicolae Anghel 《Proceedings of the American Mathematical Society》1996,124(11):3555-3561
In this paper we address the problem of generic vanishing for (negative) harmonic spinors of Dirac operators coupled with variable metric connections.
16.
Plamen Djakov 《Journal of Differential Equations》2005,210(1):178-216
Let us consider the Dirac operator
17.
Christian Bär 《Annals of Global Analysis and Geometry》1992,10(2):171-177
We derive upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds. In the case of the classical Dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature. 相似文献
18.
Helga Baum 《Annals of Global Analysis and Geometry》1994,12(1):193-209
We prove a sharp lower bound for the first positive eigenvalue of Dirac operators coupled to instantons and discuss the limit case. 相似文献
19.
Johannes Wittmann 《Mathematische Nachrichten》2019,292(7):1627-1635
Let M be a closed spin manifold and let N be a closed manifold. For maps and Riemannian metrics g on M and h on N, we consider the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index in . If M is 2‐dimensional, one gets a lower bound for the dimension of the kernel of out of this index. We investigate the question whether this lower bound is obtained for generic tupels . 相似文献
20.
The fundamental solutions of the super Dirac and Laplace operators and their natural powers are determined within the framework of Clifford analysis. 相似文献