Positive Eigenfunctions of a Schrödinger Operator

The paper considers the eigenvalue problem −∆u − αu + λg(x)u = 0 with u ∈ H(R ), u = 0, where α, λ ∈ R and g(x) ≡ 0 on Ω, g(x) ∈ (0, 1] on R \ Ω and lim |x |→+∞ g(x) = 1 for some bounded open set Ω ∈ RN . Given α > 0, does there exist a value of λ > 0 for which the problem has a positive solution? It is shown that this occurs if and only if α lies in a certain interval (Γ, ξ1) and that in this case the value of λ is unique, λ = Λ(α). The properties of the function Λ(α) are also discussed.