The $L^{p}$ boundedness of the wave operators for matrix Schr\"{o}dinger equations

TL;DR

This paper establishes the boundedness of wave operators in various L^p spaces for matrix Schrödinger equations on the line and half-line, under conditions on potential decay and scattering matrix properties.

Contribution

It proves L^p boundedness of wave operators for matrix Schrödinger equations with general boundary conditions and potentials, extending previous results to matrix and line cases.

Findings

Wave operators are bounded in L^p spaces for 1<p<∞ on the half line.

Wave operators are bounded in L^1 and L^∞ under stronger decay and scattering conditions.

Results on the line are derived from half-line cases via matrix extensions.

Abstract

We prove that the wave operators for $n \times n$ matrix Schr\"odinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(\mathbb R^+, \mathbb C^n), 1 < p < \infty, $ for slowly decaying selfadjoint matrix potentials, $V, $ that satisfy $\int_{0}^{\infty }\, (1+x) |V(x)|\, dx < \infty.$ Moreover, assuming that $\int_{0}^{\infty }\, (1+x^\gamma) |V(x)|\, dx < \infty, \gamma > \frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(\mathbb R^+, \mathbb C^n),$ and in $L^\infty(\mathbb R^+, \mathbb C^n).$ We also prove that the wave operators for $n\times n$ matrix Schr\"odinger equations on the line are bounded in the spaces $L^p(\mathbb R, \mathbb C^n), 1 < p < \infty, $ assuming that the perturbation consists of a point interaction at the origin and of a potential, $\mathcal V,$ that satisfies the condition $\int_{-^{\infty}}^{\infty}\, (1+|x|)\, |\mathcal V(x)|\, dx < \infty.$ Further, assuming that $\int_{-\infty}^{\infty }\, (1+|x|^\gamma) |\mathcal V(x)|\,dx < \infty, \gamma > \frac{5}{2},$ and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in $L^1(\mathbb R, \mathbb C^n),$ and in $L^\infty(\mathbb R, \mathbb C^n).$ We obtain our results for $n\times n$ matrix Schr\"odinger equations on the line from the results for $2n\times 2n$ matrix Schr\"odinger equations on the half line.