On the $\Theta$-function of a Riemannian manifold with boundary

Abstract:

Let $\Omega$ be a compact Riemannian manifold of dimension $n$ with smooth boundary. Let ${\lambda _1} < {\lambda _2} \leq \cdots$ be the eigenvalues of the Laplace-Beltrami operator with the boundary condition $[\partial /\partial n + \gamma ]\phi = 0$ . The associated $\Theta$-function ${\Theta _\gamma }(t) = \sum \nolimits _{n = 1}^\infty {\exp [ - {\lambda _n}t]}$ has an asymptotic expansion of the form \[ {(4\pi t)^{n/2}}{\Theta _\gamma }(t) = {a_0} + {a_1}{t^{1/2}} + {a_2}t + {a_3}{t^{3/2}} + {a_4}{t^2} + \cdots .\] The values of ${a_0}$ , ${a_1}$ are well known. We compute the coefficients ${a_2}$ and ${a_3}$ in terms of geometric invariants associated with the manifold by studying the parametrix expansion of the heat kernel $p(t,x,y)$ near the boundary. Our method is a significant refinement and improvement of the method used in [McKean-Singer, J. Differential Geometry 1 (1969), 43-69].