Generalized eigenfunctions of relativistic Schroedinger operators I

Abstract

Generalized eigenfunctions of the 3-dimensional relativistic Schrödinger operator $\sqrtΔ + V(x)$ with $|V(x)|\le C < x >^{-σ}$, $σ> 1$, are considered. We show that the generalized eigenfunctions can be expressed as the sum of plane waves and solutions to the time-independent relativistic Schrödinger equation with the radiation condition. If $σ>3$, then we can give pointwise estimates of the differences between the sums and the solutions.