Pointwise bounds on confined states in non-relativistic QED
M. Griesemer, V. Kußmaul
TL;DR
This work develops a vector-valued generalization of Kato's distributional inequality for the Pauli-Fierz model in non-relativistic QED and uses Simon's semigroup comparison to translate $L^2$-localization into pointwise bounds. By establishing $S|oldsymbol{ m psi}|\le ext{Re}( ext{sgn}oldsymbol{ m psi},Holdsymbol{ m psi})$ with $S=-oldsymbol{ riangle}+V- rac{8oldsymbol{ m ontfamily{cmr}{0} ext ontfamily{cmr}{0} rac{3}{}}} ext{(constant)}}$, the authors obtain subsolution estimates for eigenstates and, for eigenvalues below the ionization threshold $oldsymbol{ m\Sigma}$, pointwise exponential decay with rate $eta< obreakoldsymbol{\sqrt{oldsymbol{ m\Sigma}-oldsymbol{ lambda}}}$. They prove a vector-valued semigroup inequality $|e^{-tH}oldsymbol{ m psi}|\le e^{-tS}|oldsymbol{ m psi}|$, and from this derive that states with energy below threshold decay pointwise like $e^{-eta|oldsymbol{x}|}$. The approach avoids model-specific Feynman-Kac representations by relying on semigroup inequalities, providing a concise route to pointwise confinement results in non-relativistic QED and strengthening localization theory for Pauli-Fierz systems.
Abstract
Kato's well known distributional inequality for the magnetic Laplacian holds equally in the more general setting of non-relativistic quantum electrodynamics (QED), where the wave function is vector-valued and the vector potential is quantized. We give two new applications of this result: First, we show that eigenstates satisfy a subsolution estimate. Second, for general states, with energy distribution strictly below the ionization threshold, we give a short proof of pointwise exponential decay in the electronic configuration.