Perturbations of Dirac operators

We study general conditions under which the computations of the index of a perturbed Dirac operator $D_s=D+sZ$ localize to the singular set of the bundle endomorphism $Z$ in the semiclassical limit $s\to \infty$. We show how to use Witten’s method to compute the index of $D$ by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator $Z$. In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a ${\mathrm{spin}}^c$ manifold to maps between its even and odd spinor bundles.     

Quantized Dirac operators

We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Spectral Action for Scalar Perturbations of Dirac Operators

We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey–de Witt coefficients and make explicit calculations for the case of n-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes (J Geom Phys 57:121, 2006) and study the case of SU _q (2).     

Dirac operators on non-compact orbifolds

In this paper we prove that Dirac operators on non-compact almost complex, complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes’/Divergence theorem.     

Dirac operators on Lagrangian submanifolds

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge–de Rham Laplacian provided the complex structure identifies the spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.     

On Mpc-structures and symplectic Dirac operators

We prove that the kernels of the restrictions of the symplectic Dirac operator and one of the two symplectic Dirac–Dolbeault operators on natural sub-bundles of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute these kernels explicitly for complex projective spaces and show that the remaining Dirac–Dolbeault operator has infinite dimensional kernels on these finite rank sub-bundles. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabiliser of a Lagrangian subspace) in the $M{p}^c$ group and classify $G$-invariant $M{p}^c$-structures on symplectic manifolds with a $G$-action. We prove a variant of Parthasarathy’s formula for the commutator of two symplectic Dirac-type operators on general symmetric symplectic spaces.     

Distance preconditioning for lattice Dirac operators

We propose a preconditioning of the Dirac operator based on the factorisation of a predefined function related to the decay of the propagator with the distance. We show that it can improve the accuracy of correlators involving heavy quarks at large distances and accelerate the computation of light quark propagators.     

On the level order for Dirac operators

We start from the Dirac operator for the Coulomb potential and prove within first-order perturbation theory that degenerate levels split in a definite way depending on the sign of the laplacian of the perturbing potential.     

Location of eigenvalues of three-dimensional non-self-adjoint Dirac operators

Letters in Mathematical Physics - We prove the absence of eigenvalues of the three-dimensional Dirac operator with non-Hermitian potentials in unbounded regions of the complex plane under smallness...     

The index of scattering operators of Dirac equations

A new index formula of Atiyah Singer type for scattering operators is proved. The index corresponds to the vacuum polarization of the Fermion (on the Minkowski space) coupled to an external non abelian gauge field.     

Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications

We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials. In the general non-self-adjoint setting, we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb–Thirring inequalities. As physical applications, we investigate the damped wave equation and armchair graphene nanoribbons.     

L 2-index formulae for perturbed Dirac operators

The Callias index theorem is generalized from the Euclidean case to certain spin manifolds with warped ends, making use of certain index-preserving deformations.     

A toy model for higher spin Dirac operators

This paper deals with the higher spin Dirac operator Q_2,1 acting on functions taking values in an irreducible representation space for so(m) with highest weight $ (\tfrac{5} {2},\tfrac{3} {2},\tfrac{1} {2},...,\tfrac{1} {2}) $ (\tfrac{5} {2},\tfrac{3} {2},\tfrac{1} {2},...,\tfrac{1} {2}) . This operator acts as a toy model for generalizations of the classical Rarita—Schwinger equations in Clifford analysis. Polynomial null solutions for this operator are studied in particular.     

Characterization and uniqueness of distinguished self-adjoint extensions of Dirac operators

Distinguished self-adjoint extensions of Dirac operators are characterized by Nenciu and constructed by means of cut-off potentials by Wüst. In this paper it is shown that the existence and a more explicit characterization of Nenciu's self-adjoint extensions can be obtained as a consequence from results of the cut-off method, that these extensions are the same as the extensions constructed with cut-off potentials and that they are unique in some sense.On leave from Universität Zürich, Schöneberggasse 9, CH-8001 Zürich. Supported by Swiss National Science FoundationOn leave from Technische Universität Berlin, Straße des 17. Juni 135, D-1000 Berlin     

A priori estimates for the Hill and Dirac operators

The Hill operator Ty = −y″ + q′(t)y is considered in L ²(ℝ), where q ∈ L ²(0, 1) is a periodic real potential. The spectrum of T is absolutely continuous and consists of bands separated by gaps. We obtain a priori estimates of gap lengths, effective masses, and action variables for the KDV equation. In the proof of these results, the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator is used. Similar estimates for the Dirac operator are obtained.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q ². In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ ₊=P _? QP ₊, whereP _±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q ₊) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q ₊)=n?1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.