The Trace Function Expansion for Spherical Polygons

The full asymptotic expansion of the trace of the heat semi–group, tr(e−∆Ωt),where −∆Ω is the Dirichlet Laplace-Beltrami operator acting on L2(Ω) for geodesic spherical polygons Ω ⊂ S2, is derived in half–powers of t, and the coefficients determined explicitly. Let Ω be a non-empty open connected set in S = {x ∈ R : |x| = 1} with a piecewise geodesic boundary ∂Ω and interior angles γ1, · · · , γM . We refer to Ω as a geodesic spherical polygon. In this paper we study the asymptotic behaviour as t ↓ 0 of the trace function, Z(Ω, t) := tr(e−∆Ωt), (1) where −∆Ω is the Dirichlet Laplace–Beltrami operator acting on L(Ω). As well as being of interest to mathematicians, the asymptotic expansion of the heat kernel and the trace function has applications in theoretical physics. The asymptotic expansion gives a good approximation of curved space-time propagators, and in computing the Casimir energy. We refer to [3] for an extensive review of the applications of spectral geometry in theoretical physics. In [5] Chang and Dowker computed the vacuum energy on orbifold factors of spheres, where they utilised the trace expansion on spherical triangles and quadrilaterals of the 2–sphere. The asymptotic expansion that we give in this paper for geodesic spherical polygons has direct relevance to their calculations and applications. Furthermore, the trace expansion identifies the poles of the associated spectral zeta function, which is used in the calculations of functional determinants, which are of interest to both mathematicians [17] and physicists [7]. The results in Theorem 2 and Corollary 3 extends those of Kac [12] and van den Berg and Srisatkunarajah [2] from a polygon in R to a spherical polygon in S. Let {λj , φj} be the spectral resolution of −∆Ω. Then it is well–known that the Dirichlet eigenvalues form an increasing sequence, 0 < λ0 ≤ λ1 ≤ · · · , accumulating at infinity. Moreover, the heat semi–group e−∆Ωt on L(Ω) has a strictly positive C∞ heat kernel on Ω×Ω× (0,∞), see for example Chavel [6], which we denote by KΩ(θ1, θ2, t). Provided |Ω| <∞ it is well–known that, KΩ(θ1, θ2, t) = ∞ ∑ j=0 ejφj(θ1)φj(θ2), (2) which converges absolutely on compact subsets of Ω× Ω× (0,∞). 1991 Mathematics Subject Classification 35K05, 58G11.