Reproducing kernel functions of solutions to polynomial Dirac equations in the annulus of the unit ball in Rn and applications to boundary value problems

Let D := Sigma(n)(i=1) partial derivative/partial derivative xi ei be the Dirac operator in Rn and let P(X) =a(m) X-n + ... +a(1) X-1 + a(0) be a polynomial with complex coefficients. Differential equations of the form P(D)f = 0 are called polynomial Dirac equations. In this paper we consider Hilbert spaces of Clifford algebra-valued functions that satisfy such a polynomial Dirac equation in annuli of the unit ball in R-n. We determine an explicit formula for the Bergman kernel for solutions of complex polynomial Dirac equations of any degree m in the annulus of radii mu and 1 where mu is an element of vertical bar 0, 1 vertical bar. We further give formulas for the Szego kernel for solutions to polynomial Dirac equations of degree m < 3 in the annulus. This includes the Helmholtz and the Klein-Gordon equation as special cases. We further show the non-existence of the Szego kernel for solutions to polynomial Dirac equations of degree n >= 3 in the annulus. As all application we give an explicit representation formula for the solutions of the Helmholtz and the Klein-Gordon equation in the annulus in terms of integral operators that involve the explicit formulas of the Bergman kernel that we computed. (C) 2009 Elsevier Inc. All rights reserved.