Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

Eigenvalues of the basic Dirac operator on quaternion-Kähler foliations

In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kähler foliations. The limiting case is characterized by the existence of quaternion-Kähler Killing spinors. We end this paper by giving some examples.     

The continuous spectrum of the Dirac operator on noncompact Riemannian symmetric spaces of rank one

The continuous spectrum of the Dirac operator on the complex, quaternionic, and octonionic hyperbolic spaces is calculated using representation theory. It is proved that , except for the complex hyperbolic spaces with even, where .

     

On the spectrum of the two-dimensional periodic Dirac operator

We prove the absolute continuity of the Dirac operator spectrum inR ² with the scalar potential V and the vector potential A=(A₁, A₂) being periodic functions (with a common period lattice) such that V, A_j≠L _loc ^q (R ²), q>2. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 3–14, January, 1999.     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

The periodic Dirac operator is absolutely continuous

We consider the periodic Dirac operatorD inL ₂( ^d ). The magnetic potentialA and the electric potentialV are periodic. Ford=2 the absolute continuity ofD is established forA,VL _{r, loc} ,r>2; the proof is based on the estimates, obtained by the authors earlier [BSu2] for the periodic magnetic Schrödinger operatorM. Ford3 our considerations are based on the estimates forM, obtained in [So] forAC ^2d+3 . Under the same condition onA, forVC, the absolute continuity ofD, d3, is proved. ForA=0 the arguments of the paper give a new (and much simpler) proof of the main result of [D].The research was completed in the framework of the project INTAS-93-351.     

A gap in the energy spectrum of the one-dimensional Dirac operator

One considers the one-dimensional Dirac operator with a slowly oscillating potential (1) $$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ where . The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals (?∞,?¦q¦), (¦q¦, ∞). The interval (?¦q¦, ¦q¦) is free from spectrum. The operator has a simple eigenvalue only for singn C₊=sign C_?, situated either at the point (under the condition C₊>0) or at the point λ=?¦q¦ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation.     

Resolvent estimates and spectrum of the Dirac operator with periodic potential

Some estimates are given of the norm of the resolvent of the Dirac operator on ann-dimensional torus (n 2) for complex values of the quasimomentum. It is shown that the spectrum of the periodic Dirac operator with potential 3$$ " align="middle" border="0"> , >3, is absolutely continuous.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 3–22, April, 1995.     

The point spectrum of the Dirac operator on noncompact symmetric spaces

In this note, we consider the Dirac operator on a Riemannian symmetric space  of noncompact type. Using representation theory, we show that has point spectrum iff the -genus of its compact dual does not vanish. In this case, if  is irreducible, then with  odd, and  .

     

Eigenvalues of Spinc Dirac Operator

In this paper we research the lower bound of the eigenvalue of Spinc Dirac operator on the Spinc manifold. By the Weisenbock formula, we get an estimate of it, then following the idea of Th Friedrich [2] and X Zhang [6]. We get a finer estimate of it. As an application, we give a condition when the Seiberg-Witten equation only has 0 solution.     

On the absolutely continuous spectrum of block operator matrices

We prove an abstract theorem on the preservation of the absolutely continuous spectrum for block operator matrices. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Absolutely continuous spectrum of one random elliptic operator

We consider an elliptic random operator, which is the sum of the differential part and the potential. The potential considered in the paper is the same as the one in the Andersson model, however the differential part of the operator is different from the Laplace operator. We prove that such an operator has absolutely continuous spectrum on all of (0,∞).     

Extrinsic Bounds for Eigenvalues of the Dirac Operator

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.     

Prescribing eigenvalues of the Dirac operator

In this note we show that every compact spin manifold of dimension ≥3 can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.