On the $L^p$ boundedness of the Wave Operators for fourth order Schr\"odinger operators

TL;DR

This paper proves that wave operators for a fourth order Schrödinger operator in three dimensions are bounded on L^p spaces for all p between 1 and infinity, under certain decay and spectral conditions.

Contribution

It establishes the L^p boundedness of wave operators for fourth order Schrödinger operators with decaying potentials, extending known results to higher-order operators.

Findings

Wave operators are bounded on L^p for 1<p<∞.

Boundedness holds under decay and spectral assumptions.

Results apply to fourth order Schrödinger operators in three dimensions.

Abstract

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=\Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ then the wave operators $W_{\pm}= s\,-\,\lim_{t\to \pm \infty} e^{itH}e^{-itH_0}$ extend to bounded operators on $L^p(\mathbb R^3)$ for all $1<p<\infty$.