1 results
General kernel estimates of Schrödinger-type operators with unbounded diffusion terms
- Part of
- Journal: Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View
- Published online by Cambridge University Press: 19 May 2023, pp. 1-32
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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.
