Generalized eigenfunctions of relativistic Schrodinger operators I
Texas State University-San Marcos, Department of Mathematics
Generalized eigenfunctions of the 3-dimensional relativistic Schrödinger operator √-Δ+V(x) with |V(x)| ≤ C ⟨x⟩-σ, σ > 1, are considered. We construct the generalized eigenfunctions by exploiting results on the limiting absorption principle. We compute explicitly the integral kernal of (√-Δ -z)-1, z ∈ ℂ \ [0, +∞), which has nothing in common with the integral kernal of (-Δ -z)-1, but the leading term of the integral kernals of the boundary values (√-Δ -λ ∓i0)-1, λ > 0, turn out to be the same, up to a constant, as the integral kernals of the boundary values (-Δ -λ∓i0)-1. This fact enables us to show that the asymptotic behavior, as |x| → +∞, of the generalized eigenfunction of √-Δ + V(x) is equal to the sum of a plane wave and a spherical wave when σ > 3.
Relativistic Schrodinger operators, Pseudo-relativistic Hamiltonians, Generalized eigenfunctions, Riesz potentials, Radiation conditions
Umeda, T. (2006). Generalized eigenfunctions of relativistic Schrodinger operators I. <i>Electronic Journal of Differential Equations, 2006</i>(127), pp. 1-46.