|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]||it_IT