Higher order Schr\"odinger operators

TL;DR

This paper studies higher order Schrödinger operators with potential functions, establishing domain characterizations and semigroup generation in $L^p$ spaces using advanced operator theory techniques.

Contribution

It extends analysis of Schrödinger operators to higher orders with variable coefficients and unbounded potentials, providing new domain and generation results.

Findings

The $L^p$-realization of the operator is quasi sectorial.

The operator generates an analytic semigroup.

Domain characterized as intersection of bilaplacian and multiplication operator domains.

Abstract

In this paper we consider higher order Schr\"odinger operators $$\mathcal L u=Lu+Vu,$$ where $L$ denotes a fourth order operator and $V\geq 0$ a suitable potential. We initiate our analysis by considering the constant coefficients differential operator $L=\Delta^2$. Subsequently, we extend our results to more general operators $L$ featuring suitable variable coefficients. We are interested in domain characterization and generation properties of these operators in $L^p(\mathbb{R}^N)$ for $p \in (1, \infty)$. To address this problems we employ a noncommutative version of the Dore-Venni theorem due to Monniaux and Pr\"uss and we prove that the $L^p$-realization of $\mathcal L$ is quasi sectorial and, consequently, generates an analytic semigroup. Furthermore, this approach allows for a sharp characterization of the operator's domain as the intersection of the domains of the bilaplacian and the multiplication operator. The required assumptions allow to treat potentials that grow at infinity like $|x|^r$ for some $r<4$.