Vector-valued Schr\"odinger operators on $L^p$-spaces

TL;DR

This paper studies vector-valued Schrödinger operators with matrix potentials on $L^p$ spaces, proving they generate contraction semigroups under certain boundedness and ellipticity conditions.

Contribution

It establishes the generation of $C_0$-semigroups by vector-valued Schrödinger operators with nonnegative, locally bounded matrix potentials.

Findings

Operators generate contraction semigroups in $L^p$ spaces.

Properties of the generated semigroups are analyzed.

Conditions on potential and coefficients are specified.

Abstract

In this paper we consider vector-valued Schr\"odinger operators of the form $\mathrm{div}(Q\nabla u)-Vu$, where $V=(v_{ij})$ is a nonnegative locally bounded matrix-valued function and $Q$ is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential $V$, we assume an that it is pointwise accretive and that its entries are in $L^\infty_{\mathrm{loc}}(\mathbb{R}^d)$. Under these assumptions, we prove that a realization of the vector-valued Schr\"odinger operator generates a $C_0$-semigroup of contractions in $L^p(\mathbb{R}^d; \mathbb{C}^m)$. Further properties are also investigated.