03-551 Masao Hirokawa

Mathematical Addendum for ``Infrared Catastrophe for Nelson's Model" (mp_arc 03-512) (195K, Postscript) Dec 26, 03

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Abstract. In [mp_arc 03-512], the author used the kernel $a(k)$ of annihilation operator of bosons to describe and prove several useful formulas such as the pull-through formula on ground states. So, it is somewhat formal in the light of mathematics. Indeed, it is no wonder some mathematicians think the arguments by the symbolical formulas with $a(k)$ are so formal that they are incredulous of the arguments because of the well-known difficulty on the domain of $a(k)$. Thus, in this addendum, we list the difficulty in Remark 1 first. In order to treat it, we restate the formulas by using the smeared annihilation operator $a(f)$, $f \in L^{2}({\bf R}^{3})$, instead of $a(k)$ and present mathematically rigorous proofs of them. To author's best knowledge, the ways to prove them have not appeared in literatures yet. In this addendum, we use the same notations as in [mp_arc 03-512].

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