We study the spectral and scattering theory of some n-dimensional anisotropic Schrödinger operators. The characteristic of the potentials is that they admit limits at inﬁnity separately for each variable. We give a detailed analysis of the spectrum: the essential spectrum, the thresholds, a Mourre estimate, a limiting absorption principle and the absence of singularly continuous spectrum. Then the asymptotic completeness is proved and a precise description of the asymptotic states is obtained in terms of a suitable family of asymptotic operators.
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Serge Richard, Spectral and Scattering Theory for Schrödinger Operators with Cartesian Anisotropy. Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, pp. 73–111DOI 10.2977/PRIMS/1145475405