Mathematical Physics and Quantum Field Theory
Electron. J. Diff. Eqns., Conf. 04, 2000, pp. 265-288.

Exponential decay of two-body eigenfunctions: A review

P. D. Hislop

We review various results on the exponential decay of the eigenfunctions of two-body Schrödinger operators. The exponential, isotropic bound results of Slaggie and Wichmann for eigenfunctions of Schrödinger 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 are reviewed. The exponential, nonisotropic bounds of 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, with improvements due to Barbaroux, Combes, and Hislop, 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 Schrödinger and Dirac operators.

Published November November 3,, 2000.
Mathematics Subject Classifications: 81Q10.
Key words: Schrodinger operator, eigenfunction, exponential decay, Dirac operator.

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Peter D. Hislop
Department of Mathematics
University of Kentucky
Lexington, KY 490506-0027 USA

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