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Zbl 0801.35100
Robert, D.
Asymptotics for high energies of the scattering phase for second order perturbations of the Laplacian. (Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du Laplacien.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 25, No. 2, 107-134 (1992). ISSN 0012-9593

One considers a perturbation of the operator $-\Delta$ in the space $L\sb 2(\bbfR\sp d)$ by a differential operator of second order with coefficients $V\sb \alpha$ satisfying the bound $D\sp \kappa V\sb \alpha(x)= O(\vert x\vert\sp{-\rho-\vert\kappa\vert})$, $\rho> d$, $\forall\kappa$, as $\vert x\vert\to \infty$. Then the Krein spectral shift function $\xi(\lambda)$, $\lambda \in\bbfR$, is well defined.\par The goal of the paper is to find the asymptotics of $\xi(\lambda)$ as $\lambda\to\infty$. In particular, it is shown that $\xi(\lambda)= c\lambda\sp{n/2}+ O(\lambda\sp{(n-1)/2})$ with some explicit constant $c$. Furthermore, if the metric corresponding to the perturbation of order 2 does not have trapped trajectories, then the complete asymptotic expansion of $\xi'(\lambda)$ in powers of $\lambda\sp{-1}$ is given.
[D.R.Yafaev (Rennes)]

MSC 2000:: *35P25 Scattering theory (PDE)
47A55 Perturbation theory of linear operators

Keywords: Krein spectral shift function; complete asymptotic expansion

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