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 ∈ Lloc n/2(ℝn) (or V ∈ Llocr (ℝn), r > 1, if n = 2) and limR→∞ Rβ(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∥Wm|x|β(q) (Δ + 1)u∥ la(Lp(q))(ℝn) for all smooth compactly supported functions u and a suitable sequence of weights Wm, where p(q) and p′(q) are dual exponents with the property that 1/p(q) - 1/p′(q) = 1/q.
All Science Journal Classification (ASJC) codes
- Geometry and Topology