Localization of the number of photons of ground states in nonrelativistic QED

TL;DR

This paper proves that the ground state of a one-electron system coupled to a massless quantized radiation field in nonrelativistic QED is localized in photon number and exhibits exponential decay in the electron position, without infrared cutoff.

Contribution

It establishes the photon number localization and exponential decay properties of the ground state in nonrelativistic QED without infrared cutoff.

Findings

Ground state belongs to the intersection of domains of all powers of the number operator.

Photon number localization with exponential decay in electron position.

Ground state exhibits exponential decay in electron position for all photon number levels.

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 $\LR\otimes\fff\cong L^2(\BR;\fff)$, 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 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 $m\geq 0$, $D_k>0$, and $\delta>0$ 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.