 01148 P. D. Hislop
 Exponential decay of twobody eigenfunctions: A review
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Apr 20, 01

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Abstract. We review various results on the exponential decay of the
eigenfunctions of twobody 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.
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