The Asymptotic Behavior of the Principal Eigenvalue for Small Perturbations of Critical One-Dimensional Schrödinger Operators with V(X) = l <SUB>±</SUB> /x2 for ±x ≫ 1

Let H = −d²/dx² + V on R, where V(x) = l₁/x², on x ≫ 1, and V(x) = l₂/x², on x ≪ −1, for constants l₁, l₂. Assume that H is a critical operator. It turns out that it is possible to realize a critical operator H of the above form if and only if min(l₁, l₂) ≥ −¼. Denote the ground state of H by φ₀. Let W be a compactly supported function and define H_ε = H + εW. It is known that Hε will possess a negative eigenvalue for ε > 0 if and only if I = ∫_R Wφ²₀ dx ≤ 0. This negative eigenvalue, λ_ε, is unique if ε > 0 is sufficiently small. We obtain the leading order asymptotics for λ_ε, as ε → 0. In particular, the order of decay depends on whether I = 0 or I &lt; 0, and also varies continuously as min(l₁, l₂) varies in the interval [−¼, ¾]. The order of decay is independent of min(l₁, l₂), for min(l₁, l₂) > ¾, but this order is not equal to the order when min(l₁, l₂) = ¾.Copyright 1995, 1999 Academic Press, Inc.