‐Theory for Schrödinger systems

Abstract

In this article, we study for p∈(1,∞) the Lp‐realization of the vector‐valued Schrödinger operator Lu:=prefix div (Q∇u)+Vu. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the Lp‐realization of scriptL, defined on the intersection of the natural domains of the differential and multiplication operators which form scriptL, generates a strongly continuous contraction semigroup on Lp(double-struckRd;double-struckCm). We also study additional properties of the semigroup such as extension to L1, positivity, ultracontractivity and prove that the generator has compact resolvent.

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

“…An easy scalar one-dimensional counterexample shows that, even in the case when the diffusion coefficient of the operator A is constant and the drift grows slightly more than linearly at infinity, there exists no realization of the operator A in L p (ℝ) which generates a strongly continuous or an analytic semigroup (see [34]). Similar pathological behaviors are exhibited in the vector-valued case (see, e.g., [22,Example 2.2], [25,Example 2.3], [9,Sect. 3]).…”

“…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.…”

“…Unfortunately, such information for elliptic operators with unbounded coefficients is not available yet. The technique in [22] has been recently used in [25] to prove generation results in L p -spaces for vector-valued Schrödinger operators of the form Au = div(Q∇u) − Vu . In that paper, the entries of the potential V of operator A are locally Lipschitz continuous on ℝ d , satisfy the conditions ⟨V(x) , ⟩ ≥ � � 2 , for every x ∈ ℝ d , ∈ ℝ m , and |D j V(−V) − | ∈ L ∞ (ℝ d ) for some ∈ [0, 1∕2) .…”

“…Under slightly different hypotheses on the potential V (pointwise accretivity and local boundedness), generation results for the operator A as above are proved in [26], but, differently from [25], only a weak characterization of the domain is provided (in fact, the generation result is proved in the maximal domain of the realization of the operator A in L p (ℝ d ;ℝ m )).…”

“…Assuming that the diffusion coefficients are bounded, our assumptions (see Hypotheses 3.1) cover the cases considered in [22,25,26,28], where roughly speaking the coefficients may grow at most polynomially, allowing also for some exponential growth. We point out that, already in the case of Schrödinger vector-valued operators, our hypotheses allow for the entries of V to grow at infinity as e |x| , for every 𝛽 > 0 , improving the growth-rate considered in [28] (see Example 5.1).…”