A Sharp Comparison Result Concerning Schrodinger Heat Kernels

Let G₀ and G be the heat kernels of and –V respectively.Under some sharp conditions on V, a proof is given that G(x,t;y,0)/G₀(x,t;y,0)can be bounded from above and below by two positive constants.This largely improves the well-known bounds G(x,t;y,0)/G₀(x,t;y,0) t (0,T]. The result answers an open question of V. Liskevichand Yu. Semenov, in the case where V 0. A sharp global boundin time is also obtained when V 0. 2000 Mathematics SubjectClassification 35K05.