On Vector-Valued Schrodinger Operators in Lp-spaces

Abstract

... We construct a realization Ap of A in the spaces Lp(Rd;Cm), 1 p < 1, that generates a contractive strongly continuous semigroup. First, by using form methods, we obtain generation of holomorphic semigroups when the potential V is symmetric. In the general case, we use some other techniques of functional analysis and operator theory to get a m-dissipative realization. But in this case the semigroup is not, in general, analytic. We characterize the domain of the operator Ap in Lp(Rd;Cm) by using rstly a non commutative version of the Dore-Venni theorem and then a perturbation theorem due to Okazawa. We discuss some properties of the semigroup such as analyticity, compactness and positivity. We establish ultracontractivity and deduce that the semigroup is given by an integral kernel. Here, the kernel is actually a matrix whose entries satisfy Gaussian upper estimates. Further estimates of the kernel entries are given for potentials with a diagonal of polynomial growth. Suitable estimates lead to the asymptotic behavior of the eigenvalues of the matrix Schr odinger operator when the potential is symmetric. [edited by Author]