On the absence of positive eigenvalues of Schrödinger operators with rough potentials

We prove the absence of positive eigenvalues of Schrödinger operators H = -Δ + V on Euclidean spaces ℝⁿ for a certain class of rough potentials V. To describe our class of potentials fix an exponent q ∈ [n/2, ∞] (or q ∈ (1, ∞] if n = 2) and let β(q) = (2q - n)/(2g). For the potential V we assume that V ∈ L_loc ^n/2(ℝⁿ) (or V ∈ L_loc^r (ℝⁿ), r > 1, if n = 2) and lim_R→∞ R^β(q)∥V∥_{Lq(R≤|x|≤2R)} = 0. Under these assumptions we prove that the operator H does not admit positive eigenvalues. The case q = ∞ was considered by Kato [K]. The absence of positive eigenvalues follows from a uniform Carleman inequality of the form ∥W _mu∥_{la(Lp′(q))(ℝn)} ≤ C _q∥W_m|x|^β(q) (Δ + 1)u∥ _{la(Lp(q))(ℝn)} for all smooth compactly supported functions u and a suitable sequence of weights W_m, where p(q) and p′(q) are dual exponents with the property that 1/p(q) - 1/p′(q) = 1/q.