On a polynomial scalar perturbation of a Schrödinger system in $L^p$-spaces

Abstract

In the paper \cite{KLMR} the $L^p$-realization $L_p$ of the matrix Schrödinger 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 Schrödinger operator are investigated.

18 pages, no figures