Multidimensional Hardy inequalities and the absence of positive eigenvalues for the Schr�dinger operator with a complex potential

Abstract

The operator −(d/dx)2+ν where ν is a complex function, |J(x)(1+|x|1+e|>∞, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator −(d/dx)z−λ,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator −Δ+ν.