Schrödinger operators with singular potentials

Abstract

Recently B. Simon proved a remarkable theorem to the effect that the Schrödinger operatorT=−Δ+q(x) is essentially selfadjoint onC ^∞₀ (R ^m if 0≦q ∈L ²(R ^m). Here we extend the theorem to a more general case,T=−Σ ^=1/m _j (∂/∂x _j −ib _j(x))² +q ₁(x) +q ₂(x), whereb _j, q₁,q ₂ are real-valued,b _j ∈C(R ^m),q ₁ ∈L ²_loc (R ^m),q ₁(x)≧−q*(|x|) withq*(r) monotone nondecreasing inr ando(r ²) asr → ∞, andq ₂ satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq ₁ is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.