The $L^p$-boundedness of wave operators for two dimensional Schr\"odinger operators with threshold singularities

Abstract: We generalize the recent result of Erdo{u g}an, Goldberg and Green on the

L^{p}

-boundedness of wave operators for two dimensional Schr"odinger operators and prove that they are bounded in

L^{p} (R^{2})

for all

1 < p < i n f t y

if and only if the Schr"odinger operator possesses no

p

-wave threshold resonances, viz. Schr"odinger equation

(- l a p + V (x)) u (x) = 0

possesses no solutions which satisfy

u (x) = (a_{1} x_{1} + a_{2} x_{2}) | x |^{- 2} + o (| x |^{- 1})

as

| x | o i n f t y

for an

(a_{1}, a_{2}) i n R^{2} s e t m i n u s (0, 0)

and, otherwise, they are bounded in

L^{p} (R^{2})

for

1 < p l e q 2

and unbounded for

2 < p < i n f t y

. We present also a new proof for the known part of the result.

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