Lp-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

M. Burak Erdoğan; William R. Green; Kevin LaMaster

doi:10.3934/dcds.2025019

L^p-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

M. Burak Erdoğan, William R. Green, Kevin LaMaster

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the higher order Schrödinger operator H = (−∆)^m + V (x) in n dimensions with real-valued potential V when n > 4m, m ∈ N. We adapt our recent results for m > 1 to show that when H has a threshold eigenvalue the wave operators are bounded on L^p(Rⁿ) for the natural range 1 ≤ p < ₂ⁿ_m in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when m = 1. The proof applies in the classical m = 1 case as well and simplifies the argument.

Original language	English (US)
Pages (from-to)	3258-3274
Number of pages	17
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	45
Issue number	9
DOIs	https://doi.org/10.3934/dcds.2025019
State	Published - Sep 2025

Keywords

eigenvalue
higher order Schrödinger
L-boundedness
L-continuity
Wave operators

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Online availability

10.3934/dcds.2025019

Library availability

Discover UIUC Full Text

Cite this

@article{4efb943493874d5888b72b9c35940860,

title = "Lp-continuity of wave operators for higher order Schr{\"o}dinger operators with threshold eigenvalues in high dimensions",

abstract = "We consider the higher order Schr{\"o}dinger operator H = (−∆)m + V (x) in n dimensions with real-valued potential V when n > 4m, m ∈ N. We adapt our recent results for m > 1 to show that when H has a threshold eigenvalue the wave operators are bounded on Lp(Rn) for the natural range 1 ≤ p < 2nm in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when m = 1. The proof applies in the classical m = 1 case as well and simplifies the argument.",

keywords = "eigenvalue, higher order Schr{\"o}dinger, L-boundedness, L-continuity, Wave operators",

author = "Erdoğan, \{M. Burak\} and Green, \{William R.\} and Kevin LaMaster",

note = "The first and third authors were partially supported by the NSF grant DMS-2154031. The second author is partially supported by Simons Foundation Grant 511825. Partial financial support was received from the Simons Foundation Grant 511825, and the National Science Foundation grant DMS-2154031. The authors have no competing interests to declare that are relevant to the content of this article.",

year = "2025",

month = sep,

doi = "10.3934/dcds.2025019",

language = "English (US)",

volume = "45",

pages = "3258--3274",

journal = "Discrete and Continuous Dynamical Systems- Series A",

issn = "1078-0947",

publisher = "American Institute of Mathematical Sciences",

number = "9",

}

TY - JOUR

T1 - Lp-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

AU - Erdoğan, M. Burak

AU - Green, William R.

AU - LaMaster, Kevin

N1 - The first and third authors were partially supported by the NSF grant DMS-2154031. The second author is partially supported by Simons Foundation Grant 511825. Partial financial support was received from the Simons Foundation Grant 511825, and the National Science Foundation grant DMS-2154031. The authors have no competing interests to declare that are relevant to the content of this article.

PY - 2025/9

Y1 - 2025/9

N2 - We consider the higher order Schrödinger operator H = (−∆)m + V (x) in n dimensions with real-valued potential V when n > 4m, m ∈ N. We adapt our recent results for m > 1 to show that when H has a threshold eigenvalue the wave operators are bounded on Lp(Rn) for the natural range 1 ≤ p < 2nm in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when m = 1. The proof applies in the classical m = 1 case as well and simplifies the argument.

AB - We consider the higher order Schrödinger operator H = (−∆)m + V (x) in n dimensions with real-valued potential V when n > 4m, m ∈ N. We adapt our recent results for m > 1 to show that when H has a threshold eigenvalue the wave operators are bounded on Lp(Rn) for the natural range 1 ≤ p < 2nm in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when m = 1. The proof applies in the classical m = 1 case as well and simplifies the argument.

KW - eigenvalue

KW - higher order Schrödinger

KW - L-boundedness

KW - L-continuity

KW - Wave operators

UR - http://www.scopus.com/inward/record.url?scp=105002027690&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=105002027690&partnerID=8YFLogxK

U2 - 10.3934/dcds.2025019

DO - 10.3934/dcds.2025019

M3 - Article

AN - SCOPUS:105002027690

SN - 1078-0947

VL - 45

SP - 3258

EP - 3274

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 9

ER -