Heat-kernel coefficients of the Laplace operator on the 3-dimensional ball

We consider the heat-kernel expansion of the massive Laplace operator on the three dimensional ball with Dirichlet boundary conditions. Using this example, we illustrate a very effective scheme for the calculation of an (in principle) arbitrary number of heat-kernel coefficients for the case where the basis functions are known. New results for the coefficients B 5 2 , ..., B5 are presented. ∗Alexander von Humboldt foundation fellow, E-mail address: [email protected] 0 It is important to know the explicit form for the coefficients in the short-time expansion of the heat-kernel, K(t), for some Laplacian-like operator on a d-dimensional manifold M. In mathematics the interest stems for example from connections between the heatequation and the Atiyah-Singer index theorem [1], whereas in physics the interest in this expansion lies for example in the domain of quantum field theory where it is commonly known as the (integrated) Schwinger-De Witt proper-time expansion [2]. If the manifold M has a boundary ∂M, the coefficients Bn in the short time expansion have volume and boundary parts [3]. Thus K(t) ∼ (4πt)− d2 ∞ ∑ k=0,1/2,1,... Bkt k (1)