Large Time Behavior of¶Schrödinger Heat Kernels and Applications

Abstract:

We obtain global in time and qualitatively sharp bounds for the heat kernel G of the Schrödinger operator −Δ +V. The potential V satisfies V(x)∼±C/d(x)^b near infinity with b∈ (0, 2). When V≥ 0 our result can be described as follows: G is bounded from above and below by the multiples of standard Gaussians with a weight function. If b>2 then the weight is bounded between two positive {\it constants}; if b=2, the weight is bounded between two positive functions of t, d(x) and d(y), which have polynomial decay; if b<2, the weight is bounded between two positive functions of t, d(x) and d(y), which have exponential decay. Up to now satisfactory bounds for heat kernels can only be found when b>2 or b<0. An application to a semilinear elliptic problem is also given.