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 ⊂ Rd. If a is nonnegative on C∞0 ( ), then either there is W >0 such that W|u|2 dx a[u] for all C∞0 ( ;R), or there is a sequence k ∈ C∞0 ( ) and a function >0 satisfying L =0 such that a[ k] → 0, k → locally uniformly in \{x0}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in . In the latter case, one has an inequality of Poincaré type: there exists W >0 such that for every ∈ C∞0 ( ;R) satisfying dx = 0 there exists a constant C>0 such that C−1 W|u|2 dx a[u]+C| u dx|2 for all u ∈ C∞0 ( ;R). © 2005 Elsevier Inc. All rights reserved.