# The $L_{p}$ boundedness of the wave operators for matrix Schrödinger equations

### Ricardo Weder

Universidad Nacional Autónoma de México, Mexico

## Abstract

We prove that the wave operators for $n×n$ matrix Schrödinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L_{p}(R_{+},C_{n}),$ $1<p<∞,$ for slowly decaying selfadjoint matrix potentials $V$ that satisfy the condition $∫_{0}(1+x)∣V(x)∣dx<∞.$ Moreover, assuming that $∫_{0}(1+x_{γ})∣V(x)∣dx<∞,$ $γ>25 ,$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L_{1}(R_{+},C_{n})$ and in $L_{∞}(R_{+},C_{n}).$ We also prove that the wave operators for $n×n$ matrix Schrödinger equations on the line are bounded in the spaces $L_{p}(R,C_{n}),1<p<∞,$ assuming that the perturbation consists of a point interaction at the origin and of a potential $V$ that satisfies the condition $∫_{−∞}(1+∣x∣)∣V(x)∣dx<∞.$ Further, assuming that $∫_{−∞}(1+∣x∣_{γ})∣V(x)∣dx<∞,$ $γ>25 ,$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L_{1}(R,C_{n})$ and in $L_{∞}(R,C_{n}).$ We obtain our results for $n×n$ matrix Schrödinger equations on the line from the results for $2n×2n$ matrix Schrödinger equations on the half line.

## Cite this article

Ricardo Weder, The $L_{p}$ boundedness of the wave operators for matrix Schrödinger equations. J. Spectr. Theory 12 (2022), no. 2, pp. 707–744

DOI 10.4171/JST/417