Fourth‐order Schrödinger type operator with unbounded coefficients in L2(ℝN)

In this paper, we study generation results in L2(ℝN)$$ {L}^2\left({\mathbb{R}}^N\right) $$ for the fourth‐order Schrödinger type operator with unbounded coefficients of the form A=a2Δ2+V2$$ A={a}^2{\Delta}^2+{V}^2 $$ where a(x)=1+|x|α$$ a(x)=1+{\left|x\right|}^{\alpha } $$ and V=|x|β$$ V={\left|x\right|}^{\beta } $$ with α>0$$ \alpha >0 $$ and β>(α−2)+$$ \beta >{\left(\alpha -2\right)}^{+} $$. We obtain that (−A,D(A))$$ \left(-A,D(A)\right) $$ generates an analytic strongly continuous semigroup in L2(ℝN)$$ {L}^2\left({\mathbb{R}}^N\right) $$ for N≥5$$ N\ge 5 $$. Moreover, the maximal domain D(A)$$ D(A) $$ can be characterized for N>8$$ N>8 $$ by the weighted Sobolev space D2(A)={u∈H4(ℝN):V2u,|x|2α−hD4−hu∈L2(ℝN)forh=0,1,2,3,4}.$$ {D}_2(A)=\left\{u\in {H}^4\left({\mathbb{R}}^N\right):{V}^2u,{\left|x\right|}^{2\alpha -h}{D}^{4-h}u\in {L}^2\left({\mathbb{R}}^N\right)\kern0.1em \mathrm{for}\kern0.4em h=0,1,2,3,4\right\}. $$