On vector-valued Schrödinger operators with unbounded diffusion in $$L^p$$ spaces

Abstract

We prove generation results of analytic strongly continuous semigroups on $$L^p({{\mathbb {R}}}^d,{{\mathbb {R}}}^m)$$ L p ( R d , R m ) ($$1<p<\infty $$ 1 < p < ∞ ) for a class of vector-valued Schrödinger operators with unbounded coefficients. We also prove Gaussian type estimates for such semigroups.

Generation results for vector-valued elliptic operators with unbounded coefficients in $$L^p$$ spaces

“…To the best of our knowledge, this is the first paper aimed at providing a precise characterization of the domain of the infinitesimal generator of the associated semigroup, when also the diffusion coefficients of the operator A are possibly unbounded and the operator is coupled up to the first order. Indeed, the description of the domain of the generator of operator A in L p (ℝ d ;ℝ m ) has been provided only in the papers [22,25,28], but there the coefficients of the diffusion part are bounded, and in [7,8] where there is no coupling in the first-order term.…”