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

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

“…As an immediate consequence of Proposition 2, Theorem 6 and Theorem 7, we can prove the following Gaussian type estimates for the matrix-valued kernel K which generalizes Theorem 5.4 in [25]. Proof.…”

“…In this case the domain of the L p -realization of A is the intersection of the domains of the diffusion and the potential part of the operator. By perturbing V with a scalar potential v ∈ W 1,∞ loc (R d ) satisfying the condition |∇v| ≤ cv on R d for some positive constant c (in the L 2 -setting, actually even with more general matrix-valued perturbation), a larger class of potentials has been considered in [5,25], by using perturbation results due to N. Okazawa [30]. In [23], assuming again strict ellipticity and boundedness for the diffusion coefficients, pointwise accretivity and local boundedness of the potential term, the authors prove that A, endowed with its maximal domain, generates a strongly continuous semigroup in L p (R d ; C m ).…”

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.…”

“…A more general class of potentials, whose diagonal entries are polynomials of type |x| or even |x| r log(1 + |x|) as well as e |x| , for , r ≥ 1 , is considered in [28] where the operator A is perturbed by the potential vI where the function v ∈ W 1,∞ loc (ℝ d ) satisfies the condition |∇v| ≤ cv for some positive constant c. A perturbation theorem (due to Okazawa [33] and used in [28]) works for a more general diagonal perturbation of V in the L 2 -setting (see [7]), allowing for different growth rates in the diagonal entries of the potential matrix. In [7] the operator A is also perturbed by a diagonal first-order term that can grow at most linearly at infinity.…”

“…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).…”