$L^p$ boundedness of wave operators for higher order schr\"odinger operators with threshold eigenvalues

TL;DR

This paper establishes $L^p$ boundedness of wave operators for higher order Schrödinger operators with threshold eigenvalues, extending previous results to lower dimensions and providing new boundedness results at the endpoint cases.

Contribution

It adapts and extends recent results on wave operator boundedness for higher order Schrödinger operators to lower dimensions and eigenvalue conditions, including new endpoint boundedness results.

Findings

Wave operators are bounded on $L^p$ for specific $p$ ranges depending on dimension and eigenvalue conditions.

Extension of $L^p$ boundedness results to lower dimensions $2m<n extless 4m$.

New $L^ fty$ boundedness results for $n>3$ in the classical case $m=1$.

Abstract

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when $n>4m$ to lower dimensions $2m<n\leq 4m$ to show that when $H$ has a threshold eigenvalue and no resonances, the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{2n}{n-1}$ when $n$ is odd and $1\leq p<\frac{2n}{n-2}$ when $n$ is even. We further show that if the zero energy eigenfunctions are orthogonal to $x^\alpha V(x)$ for all $|\alpha|<k_0$, then the wave operators are bounded on $1\leq p<\frac{n}{2m-k_0}$ when $k_0<2m$ in all dimensions $n>2m$. The range is $p\in [1,\infty)$ and $p\in[1,\infty]$ when $k_0=2m$ and $k_0>2m$ respectively. The proofs apply in the classical $m=1$ case as well and streamlines existing arguments in the eigenvalue only case, in particular the $L^\infty(\mathbb R^n)$ boundedness is new when $n>3$.