The structure of the wave operator in four dimensions in the presence of resonances

Abstract We show that the wave operators for Schrödinger scattering in $$\mathbb {R}^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:math> have a particular form which depends on the existence of resonances. As a consequence of this form, we determine the contribution of resonances to the index of the wave operator.