On the \(L^p\) boundedness of wave operators for four-dimensional Schrödinger operators with a threshold eigenvalue

DOI10.1007/S00023-016-0534-1zbMATH Open1364.81223arXiv1606.06691OpenAlexW3121306136MaRDI QIDQ529599

Author name not available (Why is that?)

Publication date: 19 May 2017

Published in: (Search for Journal in Brave)

Abstract: Let

H = - D e l t a + V

be a Schr"odinger operator on

L^{2} (m a t h b b R^{4})

with real-valued potential

V

, and let

H_{0} = - D e l t a

. If

V

has sufficient pointwise decay, 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^{4})

for all

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

if zero is not an eigenvalue or resonance, and on

f r a c 43 < p < 4

if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on

L^{p} (m a t h b b R^{4})

for

1 l e q p l e q f r a c 43

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

i n t_{m a t h b b R^{4}} x V (x) p s i (x), d x = 0

for all zero energy eigenfunctions

p s i

, then the wave operators are bounded on

L^{p}

for

1 l e q p < i n f t y

.

Full work available at URL: https://arxiv.org/abs/1606.06691

Mathematics Subject Classification ID

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Related Items (8)

Critical Hardy-Lieb-Thirring inequalities for fourth-order operators in low dimensions ⋮ The \(L^p\)-boundedness of wave operators for four-dimensional Schrödinger operators ⋮ Title not available (Why is that?) ⋮ Counterexamples to \(L^p\) boundedness of wave operators for classical and higher order Schrödinger operators ⋮ The \(L^{p}\) boundedness of wave operators for Schrödinger operators with threshold singularities ⋮ On the \(L^p\) boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions ⋮ A remark on \(L^{p}\) -boundedness of wave operators for two dimensional Schrödinger operators ⋮ On L^p boundedness of wave operators for 4-dimensional Schrödinger operators with threshold singularities

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