The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities II. Even dimensional case

Abstract

In this paper we consider the wave operators $W_{\pm }$ for a Schrödinger operator $H$ in ${\mathbf{R}}^n$ with $n\ge 4$ even and we discuss the $L^p$ boundedness of $W_{\pm }$ assuming a suitable decay at infinity of the potential $V$. The analysis heavily depends on the singularities of the resolvent for small energy, that is if 0-energy eigenstates exist. If such eigenstates do not exist $W_{\pm }:L^p\to L^p$ are bounded for $1\le p\le \infty$ otherwise this is true for $\frac{n}{n-2}<p<\frac{n}{2}$. The extension to Sobolev space is discussed.

59 pages

References in corpus (1)

• Dispersive estimates for Schrodinger operators in dimensions one and three