The \(L^{p}\) boundedness of wave operators for Schrödinger operators with threshold singularities

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Abstract: Let

H = - D e l t a + V

be a Schr"odinger operator on

L^{2} (m a t h b b R^{n})

with real-valued potential

V

for

n > 4

and let

H_{0} = - D e l t a

. If

V

decays sufficiently, the wave operators

W_{p m} = s - l i m_{t o p m i n f t y} e^{i t H} e^{- i t H_{0}}

are known to be bounded on

L^{p} (m a t h b b R^{n})

for all

1 l e q p l e q i n f t y

if zero is not an eigenvalue, and on

1 < p < f r a c n 2

if zero is an eigenvalue. We show that these wave operators are also bounded on

L^{1} (m a t h b b R^{n})

by direct examination of the integral kernel of the leading term. Furthermore, if

i n t_{m a t h b b R^{n}} V (x) p h i (x), d x = 0

for all eigenfunctions

p h i

, then the wave operators are

L^{p}

bounded for

1 l e q p < n

. If, in addition

i n t_{m a t h b b R^{n}} x V (x) p h i (x), d x = 0

, then the wave operators are bounded for

1 l e q p < i n f t y

.

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