Let H = −Δ + V, where V is a multiplication operator by a real-valued function V(x) on Rn which is uniformly Hölder continuous and for some ϱ > 4. The relationship between existence of positive solutions, with growth conditions, of Hg = 0 and asymptotic behaviors as t → ∞ of e−th is established. Using it B. Simon's problem for H on R2 is solved.