P. D. Hislop
Exponential decay of two-body eigenfunctions: A review
(71K, LaTex 2e)

ABSTRACT.  We review various results on the exponential decay of the 
eigenfunctions of two-body Schr\"odinger operators. The 
exponential, isotropic bound results of Slaggie and 
Wichmann \cite{[SlaggieWichmann]} for eigenfunctions 
of \Schr\ operators corresponding to 
eigenvalues below the bottom of the essential spectrum are proved. 
The exponential, isotropic bounds on eigenfunctions 
for nonthreshold eigenvalues due to Froese and Herbst 
\cite{[FroeseHerbst]} are reviewed. 
The exponential, nonisotropic bounds of Agmon \cite{[Agmon]} for eigenfunctions 
corresponding to eigenvalues below the bottom of the essential spectrum 
are developed, beginning with a discussion of the Agmon metric. 
The analytic method of Combes and Thomas \cite{[CT]}, with improvements 
due to Barbaroux, Combes, and Hislop \cite{[BCH]}, 
for proving exponential decay of the resolvent, at energies 
outside of the spectrum of the operator and localized between 
two disjoint regions, are presented in detail. 
These are applied to prove the exponential 
decay of eigenfunctions corresponding 
to isolated eigenvalues of \Schr\ and Dirac operators.