On a polynomial scalar perturbation of a Schrödinger system in Lp-spaces

In the paper [10] the Lp-realization Lpof the matrix Schrödinger operator Lu=div(Q∇u)+Vu was studied. The generation of a semigroup in Lp(Rd,Cm) and characterization of the domain D(Lp) has been established. In this paper we perturb the operator Lpby a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on Lp(Rd,Cm) with domain embedded in W2,p(Rd,Cm). 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.