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Heat Kernel Asymptotic Expansion on Unbounded Domains with Polynomially Confining Potentials

In this paper we analyze the small-t asymptotic expansion of the trace of the heat kernel associated with a Laplace operator endowed with a spherically symmetric polynomially confining potential on the unbounded, d-dimensional Euclidean space. To conduct this study, the trace of the heat kernel is expressed in terms of its partially resummed form which is then represented as a Mellin-Barnes integral. A suitable contour deformation then provides, through the use of Cauchy's residue theorem, closed formulas for the coefficients of the asymptotic expansion. The general expression for the asymptotic expansion, valid for any dimension and any polynomially confining potential, is then specialized to two particular cases: the general quartic and sestic oscillator potentials.