Osaka Journal of Mathematics

Completeness of the generalized eigenfunctions for relativistic Schrödinger operators I

Dabi Wei

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Abstract

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.

Article information

Source
Osaka J. Math., Volume 44, Number 4 (2007), 851-881.

Dates
First available in Project Euclid: 7 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1199719408

Mathematical Reviews number (MathSciNet)
MR2383813

Zentralblatt MATH identifier
1161.35035

Subjects
Primary: 35P10: Completeness of eigenfunctions, eigenfunction expansions
Secondary: 81U05: $2$-body potential scattering theory [See also 34E20 for WKB methods] 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]

Citation

Wei, Dabi. Completeness of the generalized eigenfunctions for relativistic Schrödinger operators I. Osaka J. Math. 44 (2007), no. 4, 851--881. https://projecteuclid.org/euclid.ojm/1199719408


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