Characterizations of infinitesimal relative boundedness for higher order Schrödinger operators

Abstract

Let m∈N and H=(−Δ)m/2+V be a higher order Schrödinger operators in the Euclidean space Rn with V∈Lloc1(Rn). In this paper, the authors characterize the infinitesimal relative boundedness and Trudinger's subordination for H on Lp(Rn) with p∈[1,∞), via the limit behavior of the family of operators {V(λ2−Δ)−m/2}λ∈(0,∞) and a generalized Kato-type class condition. The latter is weaker than the classical Kato class condition corresponding to the case p=1. All these characterizations are new even when H=−Δ+V is a second order Schrödinger operator.