The Asymptotic Behavior of the Principal Eigenvalue for Small Perturbations of Critical One-Dimensional Schrödinger Operators with V(X) = l± /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) ≥ −14. Denote the ground state of H by φ0. Let W be a compactly supported function and define Hϵ = H + ϵW. It is known that Hϵ will possess a negative eigenvalue for ϵ > 0 if and only if I = ∫RWφ20dx ≤ 0. This negative eigenvalue, λϵ, is unique if ϵ > 0 is sufficiently small. We obtain the leading order asymptotics for λϵ, as ϵ → 0. In particular, the order of decay depends on whether I = 0 or I < 0, and also varies continuously as min(l1, l2) varies in the interval [−14, 34]. The order of decay is independent of min(l1, l2), for min(l1, l2) > 34, but this order is not equal to the order when min(l1, l2) = 34.