A Sharp Comparison Result Concerning Schrödinger Heat Kernels

Let G 0 and G be the heat kernels of Δ and Δ− V respectively. Under some sharp conditions on V , a proof is given that G ( x,t;y ,0)/ G 0 ( x,t;y ,0) can be bounded from above and below by two positive constants. This largely improves the well‐known boundsC 1 e − c 1| x − y | 2 / t⩽ G ( x,t;y ,0)/ G 0 ( x,t;y ,0) ⩽C 2 e c 2| x − y | 2 / tt ε (0, T ]. The result answers an open question of V. Liskevich and Yu. Semenov, in the case where V ⩾ 0. A sharp global bound in time is also obtained when V ⩾ 0. 2000 Mathematics Subject Classification 35K05.