Ground-state positivity, negativity, and compactness for a Schrödinger operator in RN

We treat the Schrödinger operator A=− + q(x) • on L2(RN) with the potential q :RN → [q0,∞) bounded below and satisfying some reasonable hypotheses on the growth at infinity (faster than |x|2 as |x| → ∞). We are concerned primarily with the compactness of the resolvent (A − λI )−1 of A as an operator on the Banach space X, X = f ∈ L2 RN : f/ϕ ∈ L∞ RN , f X = ess sup RN |f |/ϕ , where ϕ denotes the ground state for A. If Λ is the ground state energy for A, we show that the restricted operator (A − λI )−1 :X→X is not only bounded, but also compact for λ ∈ (−∞,Λ). In particular, the spectra of A in L2(RN) and X coincide; each eigenfunction belongs to X. As another consequence, we obtain a maximum and an anti-maximum principles. © 2006 Elsevier Inc. All rights reserved.