Generalized eigenfunctions of relativistic Schroedinger operators in two dimensions

TL;DR

This paper analyzes the generalized eigenfunctions of a two-dimensional relativistic Schrödinger operator with decaying potential, establishing their boundedness, completeness, and asymptotic behavior, including plane and spherical wave decomposition.

Contribution

It provides explicit integral kernel computations and proves boundedness, completeness, and asymptotic properties of eigenfunctions for the relativistic Schrödinger operator in two dimensions.

Findings

Eigenfunctions are bounded on specified regions in momentum space.

Eigenfunction expansion for the absolutely continuous spectrum is established.

Eigenfunctions asymptotically resemble a sum of plane and spherical waves when potential decay rate exceeds 2.

Abstract

Generalized eigenfunctions of the two-dimensional relativistic Schr\"odinger operator $H=\sqrt{-\Delta}+V(x)$ with $|V(x)|\leq C< x>^{-\sigma}$, $\sigma>3/2$, are considered. We compute the integral kernels of the boundary values $R_0^\pm(\lambda)=(\sqrt{-\Delta}-(\lambda\pm i0))^{-1}$, and prove that the generalized eigenfunctions $\phi^\pm(x,k)$ are bounded on $R_x^2\times\{k | a\leq |k|\leq b\}$, where $[a,b]\subset(0,\infty)\backslash\sigma_p(H)$, and $\sigma_p(H)$ is the set of eigenvalues of $H$. With this fact and the completeness of the wave operators, we establish the eigenfunction expansion for the absolutely continuous subspace for $H$. Finally, we show that each generalized eigenfunction is asymptotically equal to a sum of a plane wave and a spherical wave under the assumption that $\sigma>2$.