On Positive Solutions of Critical Schrِdinger Operators in Two Dimensions

Using a probabilistic approach based on the Feynman-Kac formalism and the spectral radius of the shuttle operator, we prove that two-dimensional Schrödinger operators H=-Δ+V with short-range potentials V satisfying V(x)=|x|→ ∞ 0(|x|−2(ln (|x|))−2−ϵ) for some ϵ > 0 are critical if and only if Hψ = 0 has a positive bounded distributional solution ψ. It is shown that (apart from logarithmetic refinements) our decay assumptions on V(x) as |x|→ ∞ are the best possible. This yields a complete solution of a problem posed by Simon and extends an earlier result of Murata.