Fumio Hiroshima
Photon number localization of ground states in nonrelativistic QED
(80K, latex)

ABSTRACT.  One electron system minimally coupled to a quantized radiation field 
 is considered. 
It is assumed that the quantized radiation field is {\it massless}, and 
{\it no} infrared cutoff is imposed. 
The Hamiltonian, $H$, of this system 
is defined as a self-adjoint operator acting on 
$\LR\otimes\fff$, where $\fff$ is the Boson Fock space over $L^2(\BR\times\{1,2\})$. 
It is shown that 
the ground state, $\gr$, of $H$ belongs to 
$\cap_{k=1}^\infty D(1\otimes N^k)$, 
where $N$ denotes the photon number operator of $\fff$. 
Moreover it is shown that, 
 for almost every electron position variable $x\in\BR$ and for arbitrary $k\geq 0$, 
$\|(1\otimes \N)\gr (x) \|_\fff \leq 
D_ke^{-\delta |x|^{m+1}}$ with some constants $m$, $D_k$, and $\delta$ 
independent of $k$. 
In particular $\gr\in \cap_{k=1}^\infty 
D (e^{\beta |x|^{m+1}}\otimes N^k)$ for 
$0<\beta<\delta/2$ is obtained.