L p -Boundedness of Wave Operators for¶Two Dimensional Schrödinger Operators

Let be the two dimensional Schr?dinger operator with the real valued potential V which satisfies the decay condition at infinity for . We show that the wave operators , , are bounded in for any 1<p<∞ under the condition that H has no zero bound states or zero resonance, extending the corresponding results for higher dimensions. As W _± intertwine H ₀ and the absolutely continuous part H P _ac of H : f(H)P _ac=W _± f(H ₀ )W _± ^* for any Borel function f on ℝ¹, this reduces the various L ^p -mapping properties of f(H)P _ac to those of f(H)₀), the convolution operator by the Fourier transform of the function f(ξ²). Received: 5 April 1999 / Accepted: 26 May 1999