Generation of semigroup for symmetric matrix Schr\"odinger operators in $L^p$-spaces

TL;DR

This paper proves the generation of an analytic semigroup for symmetric matrix Schrödinger operators in $L^p$-spaces, characterizing positivity and compactness, with implications for PDE analysis.

Contribution

It establishes the generation of analytic semigroups for symmetric matrix Schrödinger operators with minimal regularity assumptions, extending previous results.

Findings

Semigroup generation in $L^p$-spaces for the operator

Characterization of positivity of the semigroup

Investigation of the semigroup's compactness

Abstract

In this paper we establish generation of analytic strongly continuous semigroup in $L^p$--spaces for the symmetric matrix Schr\"odinger operator $div(Q\nabla u)-Vu$, where, for every $x\in\mathbb{R}^d$, $V(x)=(v_{ij}(x))$ is a semi-definite positive and symmetric matrix. The diffusion matrix $Q(\cdot)$ is supposed to be strongly elliptic and bounded and the potential $V$ satisfies the weak condition $v_{ij}\in L^1_{loc}(\mathbb{R}^d)$, for all $i,j\in\{1,\dots,m\}$. We also characterize positivity of the semigroup and we investigate on its compactness.