The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities

Abstract

Let $H=-\Delta +V$ be a Schrödinger operator on $L^2({\mathbb{R}}^n)$ with real-valued potential $V$ for $n>4$ and let $H_0=-\Delta$. If $V$ decays sufficiently, the wave operators $W_{\pm }=s-{\lim }_{t\to \pm \infty }{e}^{itH}{e}^{-it{H}_0}$ are known to be bounded on $L^p({\mathbb{R}}^n)$ for all $1\le p\le \infty$ if zero is not an eigenvalue, and on $1<p<\frac{n}{2}$ if zero is an eigenvalue. We show that these wave operators are also bounded on $L^1({\mathbb{R}}^n)$ by direct examination of the integral kernel of the leading term. Furthermore, if ${\int }_{{\mathbb{R}}^n}V(x)\varphi (x)dx=0$ for all eigenfunctions $\varphi$, then the wave operators are $L^p$ bounded for $1\le p<n$. If, in addition ${\int }_{{\mathbb{R}}^n}xV(x)\varphi (x)dx=0$, then the wave operators are bounded for $1\le p<\infty$.

Incorporated referee comments and updated references. To appear in Adv. Math

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• Wave Operators for Schrödinger Operators with Threshold Singuralities, Revisited
• $L^p$-boundedness of wave operators for 2D Schrödinger operators with point interactions
• On wave operators for Schrödinger operators with threshold singuralities in three dimensions
• $L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions
• Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems
• The $L^p$-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities
• Some properties of threshold eigenstates and resonant states of discrete Schrödinger operators
• On real resonances for three-dimensional Schrödinger operators with point interactions
• Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities
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• Stability for the multi-dimensional Borg--Levinson theorem of the biharmonic operator
• Direct and inverse problems for the Schrödinger operator in a three-dimensional planar waveguide