Some Negative L(p) Results for Eigenfunction Expansions Associated with Scattering Theory.

摘要

A class of eigenfunctions and related expansions is studied for Schrodinger operators with short range potentials. Results are described which relate the continuum spectra and eigenfunctions for such operators to those of the Laplacian. The continuous portion of a Schrodinger operator with short range potential is unitarily equivalent to the Laplacian. This note studies the relationship between the eigenfunction expansions of the two operators with regard to pointwise and L sup p behavior on R sup u. In particular, one will study two questions: (1) Equiconvergence: Under what conditions do the two eigenfunction expansions (using, say, spherical means) converge at the same points. In the same L sup p spaces. (2) Equisummability: What if summation methods are introduced into the expansions. Do the results change. Reprints. (jhd)