Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity
This paper answers a question, raised by \textit{J. C. Baez} and \textit{Z.-F. Zhou} [(*) J. Funct. Anal. 94, No. 2, 397-414 (1990; Zbl 0729.47009)] whether the scattering and wave operators for \[ (\square+m^ 2)\varphi+\lambda\varphi^ 3=0,\quad m>0,\quad\lambda>0, \] are analytic or not on the whole space of finite-energy data \(L^ 1_ 2(\mathbb{R}^ 3)\oplus L_ 2(\mathbb{R}^ 3)\). The answer is affirmative. This implies, which is noted in (*), that the massive \(\varphi^ 4\) theory is completely integrable. The same holds for the massless \(\varphi^ 4\) theory proved by \textit{J. C. Baez} [J. Math. Phys. 31, No. 3, 757-762 (1990; Zbl 0715.58035)].
scientific article; zbMATH DE number 54745