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We consider the Cauchy problem of the heat equation with a potential which behaves like the inverse square at infinity. In this paper we study the large time behavior of hot spots of the solutions for the Cauchy problem, by using the asymptotic behavior of the potential at the space infinity.

We consider the Schrödinger operator H = −Δ + V (|x|) with radial potential V which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of H and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel p(x, y, t) of the typefor all x, y ∈ R N and t > 0, where U is a positive harmonic function of H. Third, if U 2 is an A2 weight on R N , then we prove a lower bound of a similar type.

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