$L^p$ boundedness of wave operators for higher order schrödinger operators with threshold eigenvalues

Abstract

We consider the higher order Schrödinger 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 $m\ge 1$ when $n>4m$ to lower dimensions $2m<n\le 4m$ 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 $1\le p<\frac{2n}{n-1}$ when $n$ is odd and $1\le p<\frac{2n}{n-2}$ 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 $1\le p<\frac{n}{2m-k_0}$ 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 $n>3$.

29 pages. Updated to address referee comments, to appear in Transactions of the AMS. arXiv admin note: text overlap with arXiv:2407.07069