Some results on second-order elliptic operators with polynomially growing coefficients in $L^p$-spaces

In this paper we study minimal realizations in $L^p(\mathbb{R}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^\alpha)\Delta + b|x|^{\alpha-2}x\cdot\nabla - c |x|^{\alpha-2} - |x|^{\beta} , \quad x \in \mathbb{R}^N, \end{equation*} where $N\geq3$, $\alpha\in[0,2)$, $\beta >0$, and $b, c$ are real numbers. We use quadratic form methods to prove that $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$ admits an extension that generates an analytic $C_0-$semigroup for all $p\in(1,\infty)$. Moreover, we give conditions on the coefficients under which this extension is precisely the closure of $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$.