Asymptotic properties of the generalized eigenfunctions of relativistic Schrödinger operators

Tomio Umeda Himeji Institute of Technology

This talk is devoted to a discussion about asymptotic properties of the generalized (or “continuum”) eigenfuctions φ^{±}(x, k) of relativistic Schrödinger operators H = \sqrt{−Δ} + V (x) in R^{3} with V (x) decreasing rapidly at infinity. Based on the construction of φ^{±}(x, k) via the limiting absorption principle for H, we show that:

(i) φ^{±}(x, k) are distorted plane waves; (ii) φ^{±}(x, k) are characterized as solutions to the Lippmann-Schwinger type integral equations, of which kernels have severe singularity. We also show that φ^{±}(x, k) are the sums of plane waves and solutions to the equations (\sqrt{−Δ}−|k|)u = f with radiation conditions, and discuss regularity of φ^{±}(x, k) with respect to the variable x. Under a more restrictive condition on V, we show that φ^{±}(x, k) are asymptotically the sums of plane waves and spherical waves.