A ground state alternative for singular Schrodinger operators

Let a be a quadratic form associated with a Schr6dinger operator L = -del . (A del) + V on a domain Omega subset of R-d. If a is nonnegative on C-0(infinity)(Q), then either there is W > 0 such that integral W vertical bar u vertical bar(2) dx <= a[u] for all C-0(infinity)(Omega; R), or there is a sequence phi(k) is an element of C-0(infinity)(Omega) and a function phi > 0 satisfying L phi = 0 such that a[phi(k)] -> 0, phi(k) -> phi locally uniformly in Omega\[x(0)]. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Omega. In the latter case, one has an inequality of Poincare type: there exists W > 0 such that for every psi is an element of C-0(infinity) (Omega; R) satisfying f psi phi dx not equal 0 there exists a constant C > 0 such that C-1 f W vertical bar u vertical bar(2) dx <= a[u] + C vertical bar integral u psi dx vertical bar(2) for all u is an element of C-0(infinity)(Omega; R). (c) 2005 Elsevier Inc. All rights reserved.