Electronic Journal of Differential Equations,
Conference 04 (2000), pp. 265-288.
Title: Exponential decay of two-body eigenfunctions: A review
Author: P. D. Hislop (Univ. of Kentucky, Lexington, KY, USA)
Abstract:
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 July 12, 2000.
Math Subject Classifications: 81Q10.
Key Words: Schrodinger operator; eigenfunction; exponential decay; Dirac operator.