Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

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

TL;DR

This work establishes $L^p$-boundedness of wave operators for higher-order Schrödinger operators $H=(- abla^2)^m+V$ in dimensions $n>4m$ when zero is an eigenvalue but there are no threshold resonances or positive eigenvalues. The authors develop a unified low-energy analysis using the stationary representation of $W_+$, the symmetric resolvent identity, and Jensen–Nenciu inversion to control the zero-energy singularity, extending and simplifying the classical $m=1$ theory. Under decay and spectral assumptions on $V$, they prove that the low-energy piece $W_{low,k}$ is bounded on $L^p(R^n)$ for all $1\le p< rac{n}{2m}$ (for sufficiently large $k$), enabling dispersive-type estimates via the intertwining identity. Overall, the paper provides a parity-agnostic, streamlined framework for $L^p$-continuity of wave operators in high dimensions with threshold eigenvalues, generalizing prior results and offering a robust tool for future dispersive analyses of higher-order operators.

Abstract

We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that when $H$ has a threshold eigenvalue the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{n}{2m}$ 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.

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

TL;DR

This work establishes -boundedness of wave operators for higher-order Schrödinger operators in dimensions when zero is an eigenvalue but there are no threshold resonances or positive eigenvalues. The authors develop a unified low-energy analysis using the stationary representation of , the symmetric resolvent identity, and Jensen–Nenciu inversion to control the zero-energy singularity, extending and simplifying the classical theory. Under decay and spectral assumptions on , they prove that the low-energy piece is bounded on for all (for sufficiently large ), enabling dispersive-type estimates via the intertwining identity. Overall, the paper provides a parity-agnostic, streamlined framework for -continuity of wave operators in high dimensions with threshold eigenvalues, generalizing prior results and offering a robust tool for future dispersive analyses of higher-order operators.

Abstract

We consider the higher order Schrödinger operator in dimensions with real-valued potential when , . We adapt our recent results for to show that when has a threshold eigenvalue the wave operators are bounded on for the natural range 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 . The proof applies in the classical case as well and simplifies the argument.
Paper Structure (4 sections, 13 theorems, 91 equations)

This paper contains 4 sections, 13 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Operator Bounds
  3. Resolvent and Inverse Expansions
  4. Technical Lemmas

Key Result

Theorem 1.1

Let $n>4m\geq 4$. Assume that $|V(x)|\lesssim \langle x\rangle^{-\beta}$, where $V$ is a real-valued potential on $\mathbb R^n$ and $\beta>n+4$ when $n$ is odd and $\beta>n+3$ when $n$ is even. If $H=(-\Delta)^m+V(x)$ has an eigenvalue at zero, but no positive eigenvalues, then $W_{low,k}$ extends t

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Lemma 2.2
  • proof : Proof of Proposition \ref{['lem:low tail low d']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 11 more