The heat equation with singular coefficients

The small time asymptotics of the kernel ofe ^?tH is defined and derived for \(H = \frac{{d^2 }}{{dx^2 }} + \frac{\kappa }{{x^2 }}\) on ?¹. Lemmas on singular asymptotics in the sense of distributions are formulated and used. The results are applied to derive an index formula on ?¹.