A ground state alternative for singular Schrödinger operators

Let a be a quadratic form associated with a Schrödinger operator L=-∇·(A∇)+V on a domain Ω⊂Rd. If a is nonnegative on , then either there is W>0 such that for all , or there is a sequence and a function ϕ>0 satisfying Lϕ=0 such that a[ϕk]→0, ϕk→ϕ locally uniformly in Ω⧹{x0}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W>0 such that for every satisfying there exists a constant C>0 such that for all .