L^p-theory for Schrödinger operators perturbed by singular drift terms

Our concern is the essential m-accretivity in L^p(R^N) (1 < p < ∞, N ∈ N) of the minimal realization of second-order elliptic operator with strongly singular drift term A_p,minu = –Δu + b|x|^–2 (x∙∇)u + Vu, where b ∈ ℝ is a constant and V ∈ L^p_loc(ℝ^N\{0}) is bounded below by the inverse-square potential: V(x) ≥ c₀|x|^–2 with a new critical constant c₀ = c₀(b, p, N) ∈ ℝ. Namely, we shall generalize the result on the Schrödinger operators (that is, A_p,min with b = 0) due to Kalf, Walter, Schmincke and Simon in Edmunds and Evans (Spectral Theory and Differential Operators, The Clarendon Press, Oxford, 1987) (p = 2) and Okazawa (Jpn. J. Math. 22, 199–239, 1996) (p ∈ (1, ∞)) to that on the general case of A_p,min with b ≠ 0. The proof is based on those techniques in singular perturbation of m-accretive operators and resolvent-positivity together with Kato’s inequality. These ideas in operator theory are summarized in Okazawa (Jpn. J. Math. 22, 199–239, 1996).