Dereziński, Jan; Siemssen, Daniel An evolution equation approach to the Klein-Gordon operator on curved spacetime. (English) Zbl 1423.58016 Pure Appl. Anal. 1, No. 2, 215-261 (2019). The authors develop a theory for the Klein-Gordon equation on curved spacetimes. Precisely, they consider the Klein-Gordon operator on a Lorentzian manifold coupled to an electromagnetic potential \(A\) and a scalar potential \(Y\): \[K = |g|^{-1/2} (D_{\mu} - A_{\mu} )|g|^{1/2} g^{\mu \nu} (D_{\nu} -A_{\nu}) +Y\] where \(|g|= |\det[ g_{\mu \nu}]|\) and \(D_{\mu}= -i \partial_{\mu}\). Both classical propagators and some families of nonclassical propagators are constructed. The theory of nonautonomous evolution equations on Hilbert spaces is used to this end. Assumptions on the metric and potentials are weak, but global in spacetime. In an Appendix, the authors illustrate how some of the assumptions can be realized. Reviewer: Eric Stachura (Marietta) Cited in 17 Documents MSC: 58J45 Hyperbolic equations on manifolds 35L05 Wave equation 47D06 One-parameter semigroups and linear evolution equations 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81T20 Quantum field theory on curved space or space-time backgrounds Keywords:Klein-Gordon equation; Klein-Gordon operator; propagator; evolution equation; quantum field theory; quantum field theory in curved spacetimes PDFBibTeX XMLCite \textit{J. Dereziński} and \textit{D. Siemssen}, Pure Appl. Anal. 1, No. 2, 215--261 (2019; Zbl 1423.58016) Full Text: DOI arXiv References: [1] 10.1090/memo/1138 · Zbl 1435.81003 · doi:10.1090/memo/1138 [2] ; Bachelot, Ann. Inst. H. Poincaré Phys. Théor., 70, 41 (1999) [3] 10.4171/037 · Zbl 0228.00003 · doi:10.4171/037 [4] ; Beem, Global Lorentzian geometry. Monographs and Textbooks in Pure and Applied Mathematics, 202 (1996) · Zbl 0846.53001 [5] 10.1007/s00220-005-1346-1 · Zbl 1081.53059 · doi:10.1007/s00220-005-1346-1 [6] 10.1088/0264-9381/29/14/145001 · Zbl 1246.83025 · doi:10.1088/0264-9381/29/14/145001 [7] ; Dimock, Ann. Inst. H. Poincaré Sect. A (N.S.), 37, 93 (1982) [8] 10.1007/s11005-017-0947-x · Zbl 1374.81068 · doi:10.1007/s11005-017-0947-x [9] ; Friedlander, The wave equation on a curved space-time. Cambridge Monographs on Mathematical Physics, 2 (1975) · Zbl 0316.53021 [10] 10.1007/BF00756661 · Zbl 0419.53041 · doi:10.1007/BF00756661 [11] 10.1088/0305-4470/30/17/016 · Zbl 0913.53033 · doi:10.1088/0305-4470/30/17/016 [12] 10.1007/s00220-017-2847-4 · Zbl 1364.35362 · doi:10.1007/s00220-017-2847-4 [13] ; Häfner, Creation of fermions by rotating charged black holes. Mém. Soc. Math. Fr. (N.S.), 117 (2009) · Zbl 1213.83007 [14] 10.1007/BF02054965 · Zbl 0043.32603 · doi:10.1007/BF02054965 [15] 10.1007/s002200100540 · Zbl 0989.81081 · doi:10.1007/s002200100540 [16] 10.1007/s00220-002-0719-y · Zbl 1015.81043 · doi:10.1007/s00220-002-0719-y [17] ; Hörmander, The analysis of linear partial differential operators, III : Pseudodifferential operators. Grundlehren der mathematischen Wissenschaften, 274 (1985) · Zbl 0601.35001 [18] 10.3792/pja/1195523678 · Zbl 0104.09304 · doi:10.3792/pja/1195523678 [19] ; Kato, Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, 132 (1966) [20] ; Kato, J. Fac. Sci. Univ. Tokyo Sect. I, 17, 241 (1970) [21] 10.1007/BF01940330 · doi:10.1007/BF01940330 [22] 10.1016/0370-1573(91)90015-E · Zbl 0861.53074 · doi:10.1016/0370-1573(91)90015-E [23] 10.1007/BF01258900 · Zbl 0468.35038 · doi:10.1007/BF01258900 [24] 10.4064/dm408-0-1 · Zbl 1011.83015 · doi:10.4064/dm408-0-1 [25] 10.1007/978-1-4612-5561-1 · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 [26] 10.1007/BF02100096 · Zbl 0858.53055 · doi:10.1007/BF02100096 [27] 10.1088/1742-6596/968/1/012011 · doi:10.1088/1742-6596/968/1/012011 [28] 10.1002/mana.201500052 · Zbl 1373.47046 · doi:10.1002/mana.201500052 [29] ; Tanabe, Functional analytic methods for partial differential equations. Monographs and Textbooks in Pure and Applied Mathematics, 204 (1997) · Zbl 0867.35003 [30] 10.1142/S0129055X11004436 · Zbl 1229.35226 · doi:10.1142/S0129055X11004436 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.