01-148 P. D. Hislop

Exponential decay of two-body eigenfunctions: A review (71K, LaTex 2e) Apr 20, 01

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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.

Files: 01-148.src( 01-148.keywords , wichmann1.tex )