Abstract
We first prove that the realization
$A_{\mathrm {min}}$
of
$A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$
in
$L^2({\mathbb {R}}^d)$
with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on
$L^2({\mathbb {R}}^d)$
which coincides on
$L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$
with the minimal semigroup generated by a realization of
$A$
on
$C_b({\mathbb {R}}^d)$
. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of
$A$
and deduce some spectral properties of
$A_{\min }$
in the case of polynomially and exponentially growing diffusion and potential coefficients.
Publisher
Cambridge University Press (CUP)