Two-sided heat kernel estimates for Schrodinger operators with unbounded potential

Consider the Schr\odinger operator $L^V=-\Delta+V$ on $\R^d$,where$V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$ with some constants $c_1,c_2>0$ and a nondecreasing and strictly positive function $g:[0,\infty)\to [1,+\infty)$ that satisfies $g(2r)\le c_0 g(r)$ for all $r>0$ and $\lim_{r\to \infty} g(r)=\infty.$ Two-sided heat kernel (i.e., density function) estimatesfor the associated Schr\{o}dinger semigroup are established.

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