Nakamura, Shu Microlocal properties of scattering matrices. (English) Zbl 1346.58012 Commun. Partial Differ. Equations 41, No. 6, 894-912 (2016). Summary: We consider the scattering theory for a pair of operators \(H_0\) and \(H= H_0+ V\) on \( L^2( M, m)\), where \(M\) is a Riemannian manifold, \(H_0\) is a multiplication operator on \(M\), and \(V\) is a pseudodifferential operator of order \(\mu\), \(\mu>1\). We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödinger operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces. Cited in 5 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35P25 Scattering theory for PDEs 81U05 \(2\)-body potential quantum scattering theory Keywords:Born approximation; microlocal analysis; scattering matrix; scattering theory PDFBibTeX XMLCite \textit{S. Nakamura}, Commun. Partial Differ. Equations 41, No. 6, 894--912 (2016; Zbl 1346.58012) Full Text: DOI arXiv References: [1] DOI: 10.1007/978-3-0348-7762-6 · doi:10.1007/978-3-0348-7762-6 [2] DOI: 10.1007/BF01788912 · Zbl 0531.35062 · doi:10.1007/BF01788912 [3] DOI: 10.1142/S0129055X99000337 · Zbl 0969.47028 · doi:10.1142/S0129055X99000337 [4] DOI: 10.1007/s00220-012-1551-7 · Zbl 1264.47015 · doi:10.1007/s00220-012-1551-7 [5] DOI: 10.4171/JST/54 · Zbl 1295.81128 · doi:10.4171/JST/54 [6] DOI: 10.1016/j.jfa.2008.02.015 · Zbl 1141.47017 · doi:10.1016/j.jfa.2008.02.015 [7] Hörmander, L. (1983/1985).The Analysis of Linear Partial Differential Operators. I–IV.New York: Springer-Verlag. · Zbl 0521.35001 [8] DOI: 10.1016/0022-1236(84)90104-6 · Zbl 0568.35022 · doi:10.1016/0022-1236(84)90104-6 [9] Isozaki H., J. Fac. Sci. Univ. Tokyo Sect. IA Math 32 pp 77– (1985) [10] DOI: 10.2977/prims/1195178787 · Zbl 0611.35090 · doi:10.2977/prims/1195178787 [11] Isozaki H., Sci. Papers College Arts Sci. Univ. Tokyo 35 pp 81– (1985) [12] DOI: 10.1007/s00023-011-0141-0 · Zbl 1250.81124 · doi:10.1007/s00023-011-0141-0 [13] DOI: 10.2140/apde.2013.6.257 · Zbl 1273.35201 · doi:10.2140/apde.2013.6.257 [14] DOI: 10.1007/s002220050326 · Zbl 0953.58025 · doi:10.1007/s002220050326 [15] Jensen A., Ann. Inst. H. Poincaré Phys. Théor 41 pp 207– (1984) [16] DOI: 10.1007/BF01211163 · Zbl 0543.47041 · doi:10.1007/BF01211163 [17] DOI: 10.1007/BF01206143 · Zbl 0639.35021 · doi:10.1007/BF01206143 [18] DOI: 10.1016/S0022-1236(02)00077-0 · Zbl 1045.35059 · doi:10.1016/S0022-1236(02)00077-0 [19] DOI: 10.1090/surv/158 · doi:10.1090/surv/158 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.