Let $G_0$ and $G$ be the heat kernels of $\Delta$ and $\Delta - V$ 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 bounds $C_1 e^{-c_1 |x-y|^2/t} \le G(x, t; y, 0)/G_0(x, t; y, 0) \le C_2 e^{c_2 |x-y|^2/t}$, $t \in (0, T]$. The result answers an open question of V. Liskevich and Yu. Semenov, in the case where $V \ge 0$. A sharp global bound in time is also obtained when $V \ge 0$.