, Volume 208, Issue 1, pp 125-152

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

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Abstract:

Let be the two dimensional Schrödinger 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 1H 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 f2).

Received: 5 April 1999 / Accepted: 26 May 1999