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

TL;DR

This paper investigates the generation of analytic semigroups by a fourth-order Schrödinger operator with unbounded coefficients in high-dimensional space, characterizing its maximal domain using weighted Sobolev spaces.

Contribution

It establishes generation results for the operator with unbounded coefficients and characterizes its maximal domain in terms of weighted Sobolev spaces for dimensions greater than 8.

Findings

The operator generates an analytic semigroup in $L^2( R^N)$ for $N extgreater 4$.

Maximal domain characterized for $N extgreater 8$ using weighted Sobolev spaces.

Conditions on coefficients ensure well-posedness of the associated evolution problem.

Abstract

In this paper we study generation results in $L^2(\mathbb{R}^N)$ for the fourth order Schr\"odinger 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 $N\geq5$. Moreover, the maximal domain $D(A)$ can be characterized for $N>8$ by the weighted Sobolev space \[ D_2(A)=\{u\in H^{4}(\mathbb{R}^N)\,:\,V^{2}u\in L^{2}(\mathbb{R}^N), |x|^{2\alpha-h}D^{4-h}u\in L^{2}(\mathbb{R}^N) \text{ for } h=0,1,2,3,4\}. \]