Fourth-order Schrödinger type operator with unbounded coefficients in $L^2({\mathbb{R}}^N)$

Abstract

In this paper we study generation results in $L^2({\mathbb{R}}^N)$ for the fourth order Schrödinger type operator with unbounded coefficients of the form $$A=a^2{\Delta }^2+V^2$$ where $a(x)=1+|x{|}^{\alpha }$ and $V=|x{|}^{\beta }$ with $\alpha >0$ and $\beta >(\alpha -2{)}^{+}$. We obtain that $(-A,D(A))$ generates an analytic strongly continuous semigroup in $L^2({\mathbb{R}}^N)$ for $N\ge 5$. Moreover, the maximal domain $D(A)$ can be characterized for $N>8$ by the weighted Sobolev space \[ D_2(A)=\{u\in H^{4}(\mathbb{R}^N)\,:\,V^{2}u\in L^{2}(\mathbb{R}^N), |x|^{2α-h}D^{4-h}u\in L^{2}(\mathbb{R}^N) \text{ for } h=0,1,2,3,4\}. \]