On a polynomial scalar perturbation of a Schr\"odinger system in $L^p$-spaces

TL;DR

This paper extends the analysis of matrix Schr"odinger operators in $L^p$-spaces by adding polynomial scalar perturbations, demonstrating the preservation of semigroup properties and deriving kernel estimates and eigenvalue asymptotics.

Contribution

It introduces polynomial scalar perturbations to the matrix Schr"odinger operator and proves the resulting semigroup retains key properties, expanding understanding of such operators.

Findings

Semigroup remains strongly continuous under polynomial scalar perturbations.

Kernel estimates and eigenvalue asymptotics are established.

Properties like analyticity, compactness, positivity, and ultracontractivity are analyzed.

Abstract

In the paper \cite{KLMR} the $L^p$-realization $L_p$ of the matrix Schr\"odinger operator $\mathcal{L}u=div(Q\nabla u)+Vu$ was studied. The generation of a semigroup in $L^p(\R^d,\C^m)$ and characterization of the domain $D(L_p)$ has been established. In this paper we perturb the operator $L_p$ of by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on $L^p(\R^d,\C^m)$ with domain embedded in $W^{2,p}(\R^d,\C^m)$. We also study the analyticity, compactness, positivity and ultracontractivity of the semigroup and prove Gaussian kernel estimates. Further kernel estimates and asymptotic behaviour of eigenvalues of the matrix Schr\"odinger operator are investigated.