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

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].