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We prove first that the realization $A_{\min}$ of$A:=\mathrm{div}(Q\nabla)-V$ in $L^2(\mathbb{R}^d)$ with unbounded coefficientsgenerates a symmetric sub-Markovian and ultracontractive semigroup on$L^2(\mathbb{R}^d)$ which coincides on $L^2(\mathbb{R}^d)\capC_b(\mathbb{R}^d)$ with the minimal semigroup generated by a realization of $A$on $C_b(\mathbb{R}^d)$. Moreover, using time dependent Lyapunov functions, weprove pointwise upper bounds for the heat kernel of $A$ and deduce somespectral properties of $A_{\min}$ in the case of polynomially and exponentiallydiffusion and potential coefficients.

General Kernel estimates of Schr\"odinger type operators with unbounded diffusion terms