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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) Zbl 0801.35100

One considers a perturbation of the operator \(-\Delta\) in the space \(L_ 2(\mathbb{R}^ d)\) by a differential operator of second order with coefficients \(V_ \alpha\) satisfying the bound \(D^ \kappa V_ \alpha(x)= O(| x|^{-\rho-|\kappa|})\), \(\rho> d\), \(\forall\kappa\), as \(| x|\to \infty\). Then the Krein spectral shift function \(\xi(\lambda)\), \(\lambda \in\mathbb{R}\), is well defined.
The goal of the paper is to find the asymptotics of \(\xi(\lambda)\) as \(\lambda\to\infty\). In particular, it is shown that \(\xi(\lambda)= c\lambda^{n/2}+ O(\lambda^{(n-1)/2})\) 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 \(\lambda^{-1}\) is given.

MSC:

35P25 Scattering theory for PDEs
47A55 Perturbation theory of linear operators
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