Generalized eigenfunctions of relativistic Schrödinger operators. I

Umeda, Tomio

Electronic Journal of Differential Equations, Tome 2006 (2006).

Résumé

Summary: Generalized eigenfunctions of the 3-dimensional relativistic Schrodinger operator $\sqrt{-\Delta} + V(x)$ with $|V(x)|\le C \langle x \rangle^{{-\sigma}}, \sigma > 1$, are considered. We construct the generalized eigenfunctions by exploiting results on the limiting absorption principle. We compute explicitly the integral kernel of $(\sqrt{-\Delta}-z)^{-1}, z \in {\mathbb C}\setminus [0, +\infty)$, which has nothing in common with the integral kernel of $({-\Delta}-z)^{-1}$, but the leading term of the integral kernels of the boundary values $(\sqrt{-\Delta}-\lambda \mp i0)^{-1}, \lambda$, turn out to be the same, up to a constant, as the integral kernels of the boundary values $({-\Delta}-\lambda \mp i0)^{-1}$. This fact enables us to show that the asymptotic behavior, as $|x| \to +\infty$, of the generalized eigenfunction of $\sqrt{-\Delta} + V(x)$ is equal to the sum of a plane wave and a spherical wave when $\sigma$.

Détail

Classification : 35P99, 35S99, 47G30, 47A40
Keywords: relativistic Schrödinger operators, pseudo-relativistic Hamiltonians, generalized eigenfunctions, Riesz potentials, radiation conditions