$L^p$-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on ${\mathbb{R}}^3$

Abstract

Let $H={\Delta }^2+V$ be the fourth-order Schrödinger operator on ${\mathbb{R}}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, then it was recently shown that the wave operators $W_{\pm }(H,{\Delta }^2)$ are bounded on $L^p({\mathbb{R}}^3)$ for all $1<p<\infty$ and unbounded at the endpoints $p=1$ and $p=\infty$. This paper is to further establish the $L^p$-boundedness of $W_{\pm }(H,{\Delta }^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_{\pm }(H,{\Delta }^2)$ are bounded on $L^p({\mathbb{R}}^3)$ for all $1<p<\infty$ in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on $L^p({\mathbb{R}}^3)$ for all $1<p<3$, but not if $3\le p\le \infty$. In the third kind resonance case, we also show that $W_{\pm }(H,{\Delta }^2)$ are bounded on $L^p({\mathbb{R}}^3)$ for all $1<p<3$ and generically unbounded on $L^p({\mathbb{R}}^3)$ for any $3\le p\le \infty$. Moreover, it is also shown that $W_{\pm }(H,{\Delta }^2)$ are bounded on $L^p({\mathbb{R}}^3)$ for all $3\le p<\infty$ if in addition $H$ has the zero eigenvalue, but no $p$-wave zero resonances and all zero eigenfunctions are orthogonal to $x_i{x}_j{x}_{k}V$ in $L^2({\mathbb{R}}^3)$ for all $i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)\in {\mathbb{R}}^3$. These results describe precisely the validity of the $L^p$-boundedness of $W_{\pm }(H,{\Delta }^2)$ in ${\mathbb{R}}^3$ for all types of singularities at the zero energy threshold with some exceptions for the endpoint cases $p=1,\infty$. As an application, $L^p$-$L^q$ decay estimates are also derived for the fourth-order Schrödinger equations and Beam equations with zero resonance singularities.

50 pages, it is the final version published in JFA 2025