The \(L^p\)-boundedness of wave operators for four-dimensional Schrödinger operators

Abstract: We prove that the low energy parts of the wave operators

W_{p} m

for Schr"odinger operators

H = - l a p + V (x)

on

R^{4}

are bounded in

L^{p} (R^{4})

for

1 < p l e q 2

and are unbounded for

2 < p l e q i n f t y

if

H

has resonances at the threshold. If

H

has eigenfunctions only at the threshold, it has recently been proved that they are bounded in

L^{p} (R^{4})

for

1 l e q p < 4

in general and for

1 l e q p < i n f t y

if all threshold eigenfunctions

p h

satisfy

i n t_{R^{4}} x_{j} V (x) p h (x) d x = 0

for

1 l e q j l e q 4

. We prove in this case that they are unbounded in

L^{p} (R^{4})

for

4 < p < i n f t y

unless the latter condition is satisfied. It is long known that the high energy parts are bounded in

L^{p} (R^{4})

for all

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

and that the same holds for

W_{p} m

if

H

has no eigenfunctions nor resonances at the threshold.