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

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