Title: Maximal and minimal forms for generalized Schrodinger operator

Authors: Qingquan Deng, Yong Ding and Xiaohua Yao

Issue: Volume 63 (2014), Issue 3, 727-738


Let $0\le V\in L^1_{\loc}(\mathbb{R}^n)$ and $H=(-\Delta)^m+V$ ($m\in\mathbb{N}$) be the generalized Schr\"odinger-type operator. Then, there are two \emph{a~priori} natural nonnegative closed forms associated with the self-adjoint extension of $H$: the maximal closed form $Q_{\max}$ defined by the sum
Q_{\max}(f,f)=Q_0(f,f)+\langle V^{1/2}f,V^{1/2}f\rangle
for any $f\in W^{m,2}(\mathbb{R}^n)$ with $V^{1/2}f\in L^2(\mathbb{R}^n)$, and the minimal closed form $Q_{\min}$ defined by the form closure of $Q_{\max}$ restricted to $C_c^{\infty}(\mathbb{R}^n)$. If $m=1$, then it was shown by T.\ Kato that the maximal and minimal forms are identical, based on his famous positivity inequality. However, for $m\ge2$, the problem of the consistency seems to have no complete answer in the case of the most general locally integrable potential.

In this paper, the authors prove that $C_c^{\infty}(\mathbb{R}^n)$ is the form core of the domain $D(Q_{\max})$ for any $0\le V\in L^p_{\loc}((\mathbb{R}^n))$ with some $p$ depending on $n,m$, which greatly improves a form core result of E.\:B.\ Davies [E.\:B.\ Davies, \textit{Limits on $L^p$ regularity of self-adjoint elliptic operators}, J.\ Differential Equations \textbf{135} (1997), no. 1, 83--102] concerning all smooth nonnegative potentials. In particular, we can choose $V\in L^1_{\loc}((\mathbb{R}^n))$ (the most general locally integrable potential class) if $2m>n$. Finally, the form core result can be applied to establish the sharp bound of the kernel of the semigroup $e^{-tH}$ for $2m>n$.