On Positive Solutions of Critical Schrödinger Operators in Two Dimensions

Abstract

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.