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Generalized eigenfunctions of relativistic Schrödinger operators in two dimensions. (English) Zbl 1173.35616

Summary: This article concerns the generalized eigenfunctions of the two-dimensional relativistic Schrödinger operator \(H=\sqrt{-\Delta}+V(x)\) with \(|V(x)|\leq C\langle x\rangle^{-\sigma}, \sigma>3/2\). We compute the integral kernels of the boundary values \(\mathbb{R}_0^\pm(\lambda)=(\sqrt{-\Delta}-(\lambda\pm i0))^{-1}\), and prove that the generalized eigenfunctions \(\varphi^\pm(x,k)\) are bounded on \(\mathbb{R}_x^2\times\{k:a\leq |k|\leq b\}\), 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 \(\sigma>2\).

MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
81U05 \(2\)-body potential quantum scattering theory
47A40 Scattering theory of linear operators
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