On Fundamental Solutions in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ?{\underline{\partial}} called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ?_z{\partial_{\underline{z}}} and ?_z^f{\partial_{\underline{z}^\dagger}} which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford–Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ?E = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.     

Eigenvalue estimates for Dirac operators in geometries with torsion

On a Riemannian spin manifold (M ⁿ , g), equipped with a non-integrable geometric structure and characteristic connection ▽ ^c with parallel torsion ▽ ^c T ^c  = 0, we can introduce the Dirac operator D ^1/3, which is constructed by lifting the affine metric connection with torsion 1/3 T ^c to the spin structure. D ^1/3 is a symmetric elliptic differential operator, acting on sections of the spinor bundle and can be identified in special cases with Kostant’s cubic Dirac operator or the Dolbeault operator. For compact (M ⁿ , g), we investigate the first eigenvalue of the operator \({\left(D^{1/3} \right)^{2}}\) . As a main tool, we use Weitzenböck formulas, which express the square of the perturbed operator D ^1/3 + S by the Laplacian of a suitable spinor connection. Here, S runs through a certain class of perturbations. We apply our method to spaces of dimension 6 and 7, in particular, to nearly Kähler and nearly parallel G ₂-spaces.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L ¹-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.     

On a Chain of Harmonic and Monogenic Potentials in Euclidean Half–space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials is constructed in the upper half of Euclidean space ? ^m+1, including a higher dimensional generalization of the complex logarithmic function. Their distributional limits at the boundary ? ^m turn out to be well-known distributions such as the Dirac distribution, the Hilbert kernel, the fundamental solution of the Laplace and Dirac operators, the square root of the negative Laplace operator, and the like. It is shown how each of those potentials may be recovered from an adjacent kernel in the chain by an appropriate convolution with such a distributional limit.     

Dirac Operator with a Potential of Special Form and with the Periodic Boundary Conditions

We consider the Dirac operator on the interval [0, 1] with the periodic boundary conditions and with a continuous potential Q(x) whose diagonal is zero and which satisfies the condition Q(x) = Q^T(1?x), x ∈ [0, 1]. We establish a relationship between the spectrum of this operator and the spectra of related functional-differential operators with involution. We prove that the system of eigenfunctions of this Dirac operator has the Riesz basis property in the space L ₂ ² [0, 1].     

L'opérateur de Dirac hypoelliptique

We announce the construction of a deformation of the Dirac operator on a compact spin manifold into a hypoelliptic Dirac operator on the total space of the tangent space. This construction gives an analogue for the Dirac operator of a related deformation we already gave for the de Rham complex. For simplicity, we only explain the construction in the case of complex manifolds. We define hypoelliptic Quillen metrics, which we compare to the classical Quillen metrics. To cite this article: J.-M. Bismut, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Principal series representations and harmonic spinors

Let G be a real reductive Lie group and G/H a reductive homogeneous space. We consider Kostant's cubic Dirac operator D on G/H twisted with a finite-dimensional representation of H. Under the assumption that G and H have the same complex rank, we construct a nonzero intertwining operator from principal series representations of G into the kernel of D. The Langlands parameters of these principal series are described explicitly. In particular, we obtain an explicit integral formula for certain solutions of the cubic Dirac equation D=0 on G/H.     

The curvature tensor and the einstein equations for a four-dimensional nonholonomic distribution

The space of possible particle velocities is a four-dimensional nonholonomic distribution on a manifold of higher dimension, say, M ⁴ × ?¹. This distribution is determined by the 4-potential of the electromagnetic field. The equations of admissible (horizontal) geodesics for this distribution are the same as those of the motion of a charged particle in general relativity theory. On the distribution, a metric tensor with Lorentzian signature (+, ?, ?, ?) is defined, which gives rise to the causal structure, as in general relativity theory. Covariant differentiation (a linear connection) and the curvature tensor for this distribution are introduced. The Einstein equations are obtained from the variational principle for the scalar curvature of the distribution. It is proved that the Dirac operator for the four-dimensional distribution can be extended to functions defined on the manifold M ⁴ × S ¹, where S ¹ is the circle. For such functions, electric charges are topologically quantized.     

Note on a theorem of H. Moscovici

Let Γ be a discrete subgroup of a semisimple Lie group G such that ΓβG has a finite volume. Using a theorem of Moscovici we express the multiplicity of discrete series representations of G in the discrete spectrum of L²(ΓβG) as the L²-index of a twisted Dirac operator. This result, which extends a result of Moscovici and of the author, holds for all integrable discrete series and for infinitely many nonintegrable discrete series. In particular, up to computing L²-indices in the special rank one case, it implies the Osborne-Warner formula.     

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L₂ for small complex L_∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices.     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

The Dirac Operator of a Commuting d-Tuple

Given a commuting d-tuple T=(T₁, …, T_d) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator D_T. Significant attributes of the d-tuple are best expressed in terms of D_T, including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension d=1, 2, …) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting d-tuple, and that two Dirac operators are isomorphic iff their associated operator d-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d -contractions of finite rank. It is shown that for the subcategory of all such T that are (a) Fredholm and and (b) graded, the curvature invariant K(T) is stable under compact perturbations. We do not know if this stability persists when T is Fredholm but ungraded, although there is concrete evidence that it does.     

Riemann Boundary Value Problems for Iterated Dirac Operator on the Ball in Clifford Analysis

In this paper we consider the Riemann boundary value problem for null solutions to the iterated Dirac operator over the ball in Clifford analysis with boundary data given in $\mathbb L _{p}\left(1<p<+\infty \right)$ -space. We will use two different ways to derive its solution, one which is based on the Almansi-type decomposition theorem for null solutions to the iterated Dirac operator and a second one based on the poly-Cauchy type integral operator.     

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.     

The index of the spin Dirac operator on the weighted projective space and the reciprocity law of the Fourier-Dedekind sum

The main purpose of this paper is to give a geometric interpretation of the reciprocity law of the Fourier-Dedekind sum given by M. Beck and S. Robins. In fact, the V-index of the spin^c Dirac operator on the weighted projective space is equal to the dimension of the space of all weighted homogeneous polynomials of given degree, and this equality gives precisely the Beck-Robins reciprocity law. In this equality, the Fourier-Dedekind sums appear as the localization terms of the V-index of the spin^c Dirac operators and have a relationship to the eta invariants of lens spaces.     

Harmonic spinors on semisimple symmetric spaces

Let G/H be a semisimple symmetric space. We consider a Dirac operator D on G/H twisted by a finite dimensional H-representation. We give an explicit integral formula for certain solutions of the equation D=0. In particular, some quotients of standard principal series representations are seen to occur in the kernel of D.     

Min-max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω⊂Rⁿ, n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min-max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.     

The Dirac spectrum on manifolds with gradient conformal vector fields

We show that the Dirac operator on a spin manifold does not admit L² eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.     

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry

In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate t ^?2λ , where λ = 1, 2, . . . is the angular momentum. Our technique is to use Chandrasekhar’s separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as t ^?2λ . For the second set, in general, the solutions tend to some explicit profile at the rate t ^?2λ . The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green’s functions for time independent Schrödinger equations associated with these wave equations.