Federbush mode decomposition of \(\phi ^ 4_ 2\) (Q1819335)

This paper completes the phase cell analysis of \(\phi^ 4_ 2\) that was initiated in the original paper of \textit{G. Battle} and \textit{D. Federbush} [Ann. Phys. 142, 95-139 (1982)]. While that paper is more important with regard to the concepts that are introduced, the analysis is flawed by a serious positivity problem, which is solved in [\textit{G. A. Battle}, J. Funct. Anal. 51, 312-325 (1983; Zbl 0513.46054)] for the Battle-Federbush expansion functions. This paper applies the positivity derived in the last cited paper to prove that the Battle-Federbush cluster expansion indeed converges for the \(\phi^ 4_ 2\) model. A year after this paper appeared, new orthonormal bases of functions were discovered which have better regularity properties than the Battle- Federbush functions. A straightforward replacement of the basis neatly excises the original error in the first cited paper.