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December 2007 Completeness of the generalized eigenfunctions for relativistic Schrödinger operators I
Dabi Wei
Osaka J. Math. 44(4): 851-881 (December 2007).


Generalized eigenfunctions of the odd-dimensional ($n \geq 3$) relativistic Schrödinger operator $\sqrt{-\Delta}+V(x)$ with $|V(x)| \leq C\langle x\rangle^{-\sigma}$, $\sigma>1$, are considered. We compute the integral kernels of the boundary values $R^{\pm}(\lambda)=(\sqrt{-\Delta}-(\lambda\pm i0))^{-1}$, and prove that the generalized eigenfunctions $\varphi^{\pm}(x,k):=\varphi_0(x,k)-R^{\mp}(|k|)V\varphi_0(x,k)$ ($\varphi_0(x,k):=e^{ix \cdot k}$) are bounded for $(x,k)\in\mathbb{R}^n\times\{k\mid a\leq |k|\leq b\}$, where $[a,b]\subset(0,\infty)\setminus\sigma_p(H)$. This fact, together with the completeness of the wave operators, enables us to obtain the eigenfunction expansion for the absolutely continuous spectrum.

On considère les fonctions propres généralisées de l'opérateur relativiste de Schrödinger $\sqrt{-\Delta}+V(x)$ où $|V(x)| \leq C\langle x\rangle^{-\sigma}$ en dimension impaire ($n \geq 3$). On calcule les noyaux intégraux associés aux valeurs limites $R^{\pm}(\lambda)=(\sqrt{-\Delta}-(\lambda\pm i0))^{-1}$, et on prouve que les fonctions propres généralisées $\varphi^{\pm}(x,k):=\varphi_0(x,k)-R^{\mp}(|k|)V\varphi_0(x,k)$ ($\varphi_0(x,k):=e^{ix\cdot k}$) sont bornées pour $(x,k)\in\mathbb{R}^n\times\{k\mid a\leq |k|\leq b\}$, où $[a,b]\subset(0,\infty)\setminus\sigma_p(H)$. Ce résultat, associé à la complétude des opérateurs d'onde, nous permet d'obtenir le développement en fonction propres pour le spectre absolument continu.


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Dabi Wei. "Completeness of the generalized eigenfunctions for relativistic Schrödinger operators I." Osaka J. Math. 44 (4) 851 - 881, December 2007.


Published: December 2007
First available in Project Euclid: 7 January 2008

zbMATH: 1161.35035
MathSciNet: MR2383813

Primary: 35P10
Secondary: 47A40 , 81U05

Rights: Copyright © 2007 Osaka University and Osaka City University, Departments of Mathematics


Vol.44 • No. 4 • December 2007
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