Exponential decay of two-body eigenfunctions: A review

We review various results on the exponential decay of the eigenfunc- tions of two-body Schrodinger operators. The exponential, isotropic bound results of Slaggie and Wichmann (15) for eigenfunctions of Schrodinger 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 (5) are reviewed. The exponential, nonisotropic bounds of Agmon (1) for eigenfunctions cor- responding 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 (4), with improvements due to Barbaroux, Combes, and Hislop (2), for proving exponential decay of the resolvent, at energies outside of the spectrum of the operator and localized between two disjoint regions, is presented in detail. The results are applied to prove the exponential decay of eigenfunctions corresponding to isolated eigenvalues of Schrodinger and Dirac operators.